I've wondered this: Aside from operation costs to run each, which one is more expensive to operate from a initial cost, expected useful life and maintenance costs? That would be really helpful to know. I have no idea on this...and that's assuming you already have service set-up for each system.
IMO, if you get 10 years out of an inverter minisplit with minimal service you are doing well.
A gas shop heater will run for 15-20 years and very likely may never need service.
The mini split may be cheaper to run, but then you need to figure in replacement costs. If you hire it out that can be several thousand dollars. Divide that cost over severl years and look at your heat and AC bill and replacement. It might not justify it.
I have said it many times, you either pay the contractor, or you pay the utility, there is no free lunch. Higher efficiency equipment will save you money on your utility bill, but it will need more service. The higher efficiency equipment is probably a better way to go, less money in the long run, but within reason. I don't advocate buying the highest efficiency equipment you can find, but go one or two steps up from the basic equipment. If the minimum for an air conditioner is 14 SEER, go with the 16 SEER. You will never see a return on 20+ SEER.
I get a kick out of gas furnaces. 95% is pretty typical. Someone has a 98% furnace available. Now I will admit I don't know the pricing, but come on, it is 2%! If it is $500 more it will never payout in the long run. I pay around $800 for natural gas a heating season. That comes to $16 a year savings. It would take 30 years to come out on it assuming $500 more for the equipment.