DIY is a real possibility now with mini-split systems. They come with pre-charged linesets, you don't need a license to install them.
If you can live with the swamp cooler for a few more years, I would. That's just because new units with hydrocarbon refrigerants are starting to roll out, R134a and R410a are being phased out, and with hydrocarbons there won't be any need to worry about discharges for service in the future, the EPA has exempted the new refrigerants from licensing requirements. The hydrocarbon refrigerant units are still expensive right now, that should change soon.
A heat pump is just an air conditioner with a reversing valve. There's no excuse for the current contractor price gouging. My 3 ton heat pump was about $3500 over a decade ago.