Is there such a thing as a DIY heat pump or mini split
Yes, there is. The problem is that a mini-split is, after all, split.  That means the pipes between compressor and evaporator are custom fit and routed through your house.  Obviously, you can't fill the pipes until after they are fit.  So if the fluid is freon, yeah, you're gonna need a license.   

However that's not the only way to do it.  

Entire Freon cycle outside

In this scenario, we have a complete unit that sits on a pad outside. It contains all the Freon machinery, pre-filled from the factory.  It makes cold water, which is piped into your house and run through a radiator (one made to deal with condensate). Then the warm water goes back out to be chilled again.  You can have as many registers as you want. 

This is called an "air to water heat pump system".

Instead of water, you use antifreeze, not for in-use times, but for the winter so you don't have to drain it. 



Outside unit, containing all the freon. The inside unit is simple.

Split Freon cycle

This is the mini-split you are familiar with. Put the Freon condenser outside and the evaporator outside, with custom fit plumbing between carrying the freon. Freon must be added, so must be installed by pro's. 

Entire Freon cycle inside

This is a factory-built box, with the Freon pre-charged, that sits inside the room being cooled.  The box gives cold air and hot water.  You plumb the hot water outside, let it cool through passive radiators, and return it to be heated again. 

Science fiction? No, there's one of these 10 feet away from me right now. 

These systems are called "water-sourced heat pumps". In a small residential system you would use antifreeze, so it doesn't break pipes in winter.



Inside unit, containing all the freon. The outside unit is simple. This thing is loud, by the way.

Most of these systems can actually reverse to be heat pumps.  In that case, when it is too cold outside for the heat pump to work, one choice is to bypass the outside radiator and run it through a fuel-fired water heater instead. That's cheaper than resistive electric "emergency heat".  Or you could interchange with groundwater, which is always at a moderate temperature. 

Alternate refrigerants

Manufacturers are starting to tinker with "alternate" refrigerants, probably to get around the environmental hazard and licensure of CHC/FHC/CFC Freons. One promotes "environmentally friendly CO2"... amusing, but that actually is reasonable since it will be captured inside a coolant loop for the next 20 years, not emitted on a regular basis. 

Amazingly, they're re-branding CO2 as R-744.  Next thing you know, they'll patent it and we'll have to pay a license fee to exhale!