I don't understand your reference to AC/Dc for fridge. A lithium battery is not going to solve your problems, it's still just a 100ah battery from which you can draw a bit more power than an acid.
Solar is a really good idea for this problem. In fact, it's about the only idea unless you like hookups all the time or running a generator.
Here's a few previous posts on the subject:
pros-cons v. gas2016boondock with residential refer 2018household refer and inverter2009 Household refer Class A 2009residential fridge- boondocking 09Residential refer Class A 2010
