Where are you going to find a mini-split that only draws 4 amps? I installed the smallest one I could find (12,000 BTU) in my cabin and it draws about 6.5 amps at 120 volts.
Even if you're unit draws 4 amps that's basically 500 watts. Overnight would be about 12 hours so you need 6,000 watt hours of storage. Using something like a Simpliphi PHI-3.8 you would still need 2 of them to just barely have enough. That's around $6,000 for the batteries. An inverter will cost you another $1,000 to $2,000. Then you need cables, lugs, etc. I'd say figure $8,000 minimum for just the inverter and batteries. If you can get 1,000 watts of solar panels on the roof that would be quite helpful. That's about another $1,000 for the panels, brackets, and wire. A charge controller will be another $500 to $1000.
I, like most people, would love to have this in an RV but the space it takes up, the weight it adds, and the money it costs are all prohibitive for most people. You can find a few Rvers on Youtube and where ever that have installed systems like this.
