We have a two-story home half his size, but one minisplit upstairs and one downstairs would be $3-4000 installed, combined, and that's upper or top-end models.
The ones I got would easily handle a house of his size.
Installed prices in the US are pretty silly. Theres a bourgeoning set of DIY units that come precharged. But even those are pricey compared to the rest of the world.
A vacuum pump costs about $100. Just get a regular cheap mini split and vacuum the lines yourself, and save $1,000 or more.
One of the interesting things I noticed here is that manufacturers are providing exchangers and head units that can reuse existing pipes. Makes sense for retrofit/upgrades where running pipes through utility spaces and bulkheads is a significant labor cost.
If I were to have it installed, just about any "professional" will totally overcharge here in the US too.
However, if I could have them installed for $600, I wouldn't bother doing it myself. We have a little bit of a cartel behavior with AC installers at least in my area of the US. Even though the job has gotten much easier with mini splits being requested more, they're unwilling to give up the amount they were making per job with Central units.
A fine I presume, if someone figures out. You'd have to get the unit second hand or similar, all stores are required to ensure you've ordered an installation from a certified company when they sell it AFAIK. Last ones I bought, a few years apart, had it all integrated in the web shops. Just selected installation company from a dropdown, fixed price for basic install.
The new heat pump system cost roughly $14,500 and provides heating and cooling via ductwork that was already in place.
(In a more enlightened world, I think that a system with the refrigerant fully contained within the outdoor unit that uses plain water as the heat exchange fluid could make a lot of sense, and this would not change the pipe situation.)