Forced air cooling is going to require duct work. Mini splits are going to require more equipment.

A budget of 5k isn't realistic with either option mentioned above for whole house solution of this size and a climate facing 100+ degrees. Certainly not no where close to what would be considered "best".

3 stories present even more challenges. Why? heat rises and thermostat wars being fought from people on 3rd floor versus those on 1st floor. (you get what you pay for here - no way around that in a hot climate)

The advantage to mini split is you could do the hottest part of the house first or part of the house you currently use the most and then add more to it later. But mini split will cost you far more in the long run due to maintenance, break and repair and ultimate replacement. Typically part warranties for those aren't as long as a traditional ducted HVAC system.

A traditional forced air duct system for an already built home 3 stories professionally done is going to cost more than 5k just by itself. Going forward this is probably the most ecconomical way in the long run via less maintenance, less repairs because there's less equipment... but you will pay for this in the way of labor / not to mention skill (hopefully). From there you have the cost of the equipment which for a 3 story home could lead to other costs like HVAC zoning either by means of controls OR even more equipment

You may be able to cut this cost some by going to high velocity system as the ducts are much smaller easier to route thru out the house. SpacePak is a common brand and there is one other in this space but the brand name escapes me right now.

Additional hurdles: Current equipment uses R410a refrigerant and is the early innings of being phased down and out. 10% cut to production now with an additional cut of 30% to production beginning next year for a total of 40% cut to production. Supply and demand determines price. The HVAC market is mostly R410a now. If everyone wants it and supply drops (it will) prices go up.

New equipment that uses new refrigerants -- isn't quite here yet. So new we don't currently know what those costs will be... info is starting to trickle out so maybe a few months from now we may have a bigger picture. Costs have done nothing but go up since the pandemic started so not likely to be cheaper. New machines with new refrigerants will likely need additional controls due to mildly flammable aspect of the new refrigerants. Additional controls means more money in layman's terms.

This Freon Wars saga is all back page news: If you want to know why this is happening? I've been covering this story for over 2 years now... from the very beginning. You can watch the currently 5 part video below. I break it down for you like no one. I plan to update this topic on an annual basis as more info trickles out.

I service the Katy, Texas area.

I a word, "No". There's no way you're going to retrofit cooling ductwork or multiple minis in a home that wasn't designed for it for $5000. Maybe $20,000, if you're lucky!

I agree, that 20, 000 might be lucky. The advantage is that you can get a heat and air unit, that will be efficient, and will save a lot of money over, an old type that is now in the house for heating. You may want to figure all that cost in, before you make an offer, if you havent already. It would be best to have a good reputable hvac person give you a complete estimate.

Short comment -

Furnaces heat air. Boilers heat water. If you have radiators, what you have is a boiler, not a furnace.

Installing air conditioning in a three story structure with no existing ducts is going to be a challenge. Probably a minisplit systems will be the best approach. I agree to be thinking of $20K if you're lucky, likely more. I wonder if the electrical system in a 100 year old structure will be able to handle that. New rewiring and new panels may be required too. Add 5 K for that.

Mini splits are a good option to install air conditioning in a house built in the 1920s. But $5K is barely enough money for to cover the costs for one floor.