"We've had a bad experience with the Carrier Infinity (see first paragraph), so are hesitant."

You had a bad experience with the installer, not the Carrier equipment. I'd rather have the cheapest Goodman system that's installer properly than the most expensive Carrier system, installed poorly. Having said that, I'd rather have the most expensive Carrier system installed properly!

I like Rheem but I don't think it's any better than Carrier. Not that familiar with Bosch but wait five minutes and someone will chip in.

Are you talking about the Carrier 25VNA8 heat pump? If you look at the specs and installation instructions (pages 6-7), you see that this heat pump can be controlled by a standard two-stage thermostat. However, this heat pump is probably significantly more expensive than a two-stage model such as the 25TPB7 so there's probably no reason to buy it and then dumb it down with a less-capable thermostat.

I have a package heat pump and I will say that it's worth a bump in price for a two-stage unit over a single stage. The 1st stage allows a slower fan speed, less "whoosh" noise from the registers, and better humidification. My heat pump virtually never runs in stage 2 even in 100-degree weather.

If you go with a two-stage HVAC system, I recommend the Honeywell T10 Pro thermostat and, if needed, the remote sensor. There can be a significant difference in temperature between my first floor and second floors so I programmed the thermostat to control the temperature based on that of the 1st floor during the day and 2nd floor (bedroom floor) at night.

Bosch Inverter is hard to beat in terms of what it does how it does it.

A communication system devised of proprietary controls leaves much to be desired as that unit ages. Carrier infinity runs via 5 speeds, so while it's inverter driven it's hindered to ramping up or down via a wider range of comfort algorithm. The carrier infinity was designed to be run from their own proprietary control to do differently than that would not be recommended.

The Bosch Inverter only needs a simple 24v signal. A single signal, not a two stage signal or any other. The Bosch inverter depending on configuration can have upwards of around 70 speeds it will run in from a single 24v signal. By having the algorithm make decisions from the outside condenser via pressure transducers, air temp sensor etc. (essentially from start up of the Bosch around 20% capacity range from there up and down 1% capacity ramps.)

Air conditioners (all of them) provide some dehumidification. Some designs do a better job of this than others. Carrier Infinity have a dehumidification setting from the thermostat but the way an air conditioner "controls" temperature the dehumidification settings are a battle from over cooling to try to dehumidify.

Usually dehumidification is controlled from a t-stat with a dehumid range? Yes. Say your set point for cooling is 70F. Dehumid is: 50% The tstat allows the unit to run for an additional 3F lower to "try" to satisfy the dehumid setting to 50%. If temp drops to 67F and dehumid setting is not reached? no bueno.

Bosch would be similar to this, but you simply control it via temp, you don't use dehumid settings. (if control of humidity is that important like you want 40% RH all the time then you need to spend more money for a dehumidifier.) For me Bosch does the job on it's own. I sit here this morning with 38% RH in my home office.

If you have a 5 speed AC that compressor is going to be similar to Bosch when trying to dehumidify. The run stage Bosch starts in around 20-25% of capacity of the machine. Carrier infinity 1/5 of capacity is? about 20%.

So the difference is made up? Between Bosch & Infinity when you need only 1% steps up and down vs 20% steps at a time for the Infinity. (Would you notice the difference between the two if both were properly installed? probably not.) Think run time thru out the day: if you need exactly 55% capacity of the machine, The properly installed Bosch will give you that. The carrier infinity? somewhere between 40 and 60% capacity. If it goes 60% it over cools and probably ends the cycle. If the cycle ends (unit shuts off) no more dehumidification until it restarts.

The infinity is more expensive. I wouldn't consider the Bosch Inverter cheap. I installed one at my home in October 2019. More importantly I would do it again. (Is Bosch perfect? no. But hard to beat for what it does / how it does it. Plus operationally speaking it saves money.)

Rheem Classic Plus: another inverter driven unit limited to 3 speeds.

• PlusOne® Variable Speed Twin Rotary Compressor & Inverter Drive: Features variable speed operation from 40 to 70% capacity, or 100% capacity via line voltage, with the EcoNet® Smart Thermostat. 3-speed operation when installed with a 24V two-stage thermostat. Provides precise temperature control, advanced humidity control and greater efficiency.

So the difference here from the first two is that start up capacity is about double the other two. So more likely to over cool with a start up of 40% capacity.

All of these machines should be ONLY serviced by Professionally Licensed HVAC contractors in your area.

I service the Katy, Texas area.

Where are you located? What is the size (sq feet) of the upstairs?

Did any of the bidders do a load calculation? I guess that matters less if you're getting variable speed equipment but you don't want to oversize it anyway.