It is difficult for existing (not to mention allowed) refrigerants to operate over a wide range of temperatures, performing heat pump functions in both directions. When the coefficient of performance gets to 1.0, you have electric heat. For any particular COP function with temperature, electric power price, and oil heating price, there is some temperature where switching from mini-split heating to oil heating is cost effective. (Also gas heating if one is so connected.)

Below are two search results related to this question. They probably address different systems.

From: https://www.greenbuildingadvisor.com/question/need-to-know-mitsubishi-heat-pump-cops-at-temps-below-17-degrees-f

My article, "How To Buy a Ductless Minisplit," included this information on Mitsubishi and Fujitsu cold-climate heat pumps:

"The equipment described in this article should perform at the following COPs:
• a COP of about 3.2 to 3.7 at 30°F;
• a COP of about 1.8 to 2.8 at 5°F;
• a COP of about 1.4 at -10°F;
• a COP of a little above 1.0 at -15°F."

From: https://aircondlounge.com/what-is-the-cop-of-heat-pumps/ [click to expand]

Forgot to add that in this mode, the exterior unit is operating even colder than ambient at the coils, so icing from air humidity is likely unless the dew point is below the coil temperature. Split minis cycle a defrosting mode, but this certainly has a lower air temperature limit for successful operation. Conceivably, coil heating with electric power could be used such that the overall COP was well under 1.0, but in that case it would be more efficient to just add interior electric heating units as backups, or a propane heater. Consider including a means of keeping pipes from freezing if inconvenient combinations of failures occur.

Here's some general information on the better quality minis:

https://www.nrel.gov/docs/fy11osti/52175.pdf

Minisplits would be a great solution if it weren't for one thing...... They cost about 4 times as much in the US as they do in the rest of the world

it seems i have only found the lower end brands that are rated for this type of cold (blueridge at -22f being the best I’ve found)

I took a look at the Blueridge mini split specification data for the BMHH48M22C outdoor model. This is a nominal 4 ton model, I could not find the 2 ton equivalent. The rating heating capacity of 50,000 BTU. At -4F degree is drops to 42,000 BTU. The capacity for -13F and -22F is blank. The outdoor unit may "operate" at -22F but you have no idea how much heat it will produce. I recommend you stay away from anything made in China.

The report in the above link is from 11 years ago. I don't doubt its accuracy, but 11 years can be a couple of generations when a technology is evolving quickly. Mitsubishi introduced their H2i series in 2014 which performs well at low temperatures. A look at more recent performance data will enable a more informed purchase decision.

To expand a bit and offer a redirect about what kaseki said, COP is a factor used to compare the output of different equipment models producing heat by electricity only. It compares heat produced (in BTUs) using a given amount of electricity, by a means other than electrical resistance, to the equivalent produced by the same amount of electricity using electrical resistance. Glowing resistance strips, like in a toaster. The scale starts at 1.0, that's the arbitrary number used as a starting point for comparison, COP of electrical resistance heating. Any number could have been chosen.

Heat pump output diminishes for a given input of electricity as outdoor temps drop (because at the lower temp there's less heat available to "extract" to pump indoors). COP is a relative comparison, not an absolute one.

The normal function of a heat pump, whether a split or a minisplit, doesn't use electrical resistance and is more efficient. A COP of 3 means that at that given outside temperature, the heat pump will produce 3 times as much heat as would be produced by the same amount of electricity powering electrical resistance strips.

Sorta. Par. 3 is correct, making allowance for marginal terminology.

In par. 1, there is nothing arbitrary about the scale: the value is heat flow rate delivered divided by the electrical power used. Heat pumps operating in their intended range can move more heat from a source to a sink than the electrical energy that they use to do so. Hence COP is usually greater than 1.0. Real heat pumps are not as efficient as theoretical heat pumps. It is possible for the COP value to be lower than 1.0 when operating well away from the range that the selected refrigerant can support. Range in this case meaning high and low temperature values.

While there is somewhat less energy available at lower temperatures, the bigger issue with efficiency is that the limit for a heat engine is related to the difference of temperatures (relative to absolute zero). Cooling from 95F to 75F will be a lot more efficient for a given refrigerant than heating from -15F to 75F.

More insight (or confusion) may be found in this summary: https://en.wikipedia.org/wiki/Heat_pump_and_refrigeration_cycle. Note equations at the end of the article.

"In par. 1, there is nothing arbitrary about the scale: the value is heat flow rate delivered divided by the electrical power used."

Okay, then work this along to show how your suggestion applies.

1000 watts of electricity used by a resistance mechanism produces 3412 BTUs. How do you apply math to these numbers to get to 1.0 COE? Unless you're dividing the heat produced in watts to the heat input in watts. But since this is an index to be used for comparisons, it could also have been set to 100 (as is often done with other indices used for comparisons) which was my comment about it being arbitrary.

"Unless you're dividing the heat [flow rate] produced in watts to the heat [flow rate] input in watts." Yes.

In the case of mini-splits, the heat produced into the heated room divided by the heat value of the electrical power used is the COP. Consistent units have to be used for both numerator and denominator: viz., BTUh, watts, joules/hr, or whatever.

I only skimmed @sktn77a's referenced document for COP values, but I think that document has its study methodology described.

I wouldn't call COP an index; it is a means of characterizing the merit of a heat engine (in this case using a refrigerant gas in liquid and vapor form) in lieu of using direct electrical heating. I think the initial users of this term were unwilling to call it an efficiency because of the generally greater than 1 characteristic, but the engine does not violate the laws of thermodynamics. The heat was taken from the outside air. Total heat moved inside is less than the sum of the heat taken from the outside air plus the heat value of the electricity used.

In general, because the choice of operating conditions (pressures, heat exchanger design, etc.) and refrigerant determine COP vs. temperature, the mini-split performance will vary over the seasons and the important comparison between mini-splits will be the relative total power used over an entire heating and cooling year. There is a value generally supplied for this, but it applies to a standard environment, and may not apply to one's particular local climate.

"I wouldn't call COP an index"

Then you don't understand what an index is and how they are used.

Using 100 is as valid (and is much more customary) to use for indices as using 1.0. Both numbers express the same precision and have the same number of significant digits. Are you going to say that you wouldn't call 100 and 1.0 as having the same number of significant digits?

from dictionary.com

index: noun. ...

• (in a nonfiction book, monograph, etc.) a more or less detailed alphabetical listing of names, places, and topics along with the numbers of the pages on which they are mentioned or discussed, usually included in or constituting the back matter.
• a sequential arrangement of material, especially in alphabetical or numerical order.
• something used or serving to point out; a sign, token, or indication: a true index of his character.
• something that directs attention to some fact, condition, etc.; a guiding principle.

COP is a power ratio. I fear we speak English differently.

If you need a dictionary, my friend, you don't understand what they are nor when and how they're used.

I researched this a few years ago, following links provided in the Efficiency Maine website giving the story of a heat pump (not mini split) installation in Presque Isle, ME which worked well. A quick Google search gave me this link: Heat Pump Systems for Maine Homes - High Efficiency Heating in Cold Climates (efficiencymaine.com). (I don't know if this is the same link which I used several years ago).

Cold climate heat pump technology is rapidly evolving. The U.S. Dept. of Energy is working with a number of manufacturers to improve efficiency as part of their "Cold Climate Heat Pump Challenge."

https://www.energy.gov/articles/doe-announces-breakthrough-residential-cold-climate-heat-pump-technology#:~:text=%E2%80%9CDOE's%20Cold%20Climate%20Heat%20Pump,each%20year%20on%20utility%20bills.

Here is the Residential Cold-Climate Heat Pump Technology Challenge Fact Sheet. It is possible to build a heat pump they achieves all the minimum requirements. The question becomes what will be the cost to the consumer. I do hope when heat pump manufacturers make claims of operating down to -22F degrees the DOE will force them to state exactly what that means in terms of providing heat output.

From the "fact sheet" linked by @mike_home it seems that the goal of a COP between 2.1 and 2.4 at +5F might be useful in Zone 5 (where I live in S. NH), depending on the price and availability of oil or gas for heating, but may not be sufficient for the OP's need in Wisconsin. In N. NH, there are examples where people have used heat pumps with the cold side heat exchanger sunk into a well (or using well water brought to the heat exchanger), thereby making use of a roughly 40F source of heat. Even with pumping costs this should yield a better COP than depending on existing refrigerants and ambient air. Such a system might be a better option for the OP, given reliable electricity.

Ref: https://www.energy.gov/energysaver/geothermal-heat-pumps

There are states which have begun to ban natural gas in new construction. The goal is to only allow all electric homes. Someday we will see heat pumps installed in new homes in the coldest parts of the U.S. This is what is driving the new heat pump technology. Add on top of this the requirement of refrigerants with low global warming potential and begin to realize the design and testing task needed to implement this in the next few years.