Good on ya for facing reality, especially since your budget is that limited.
Spend some time learning how to semi-DIY LFP while saving up for it.
Since you need to run the genny regularly anyway, that chemistry will let you capture the charging output so much faster, that as you build up your AH capacity you will directly reduce expensive runtime.
You may not ever get big enough to be able to run aircon off the bank for more than a few hours at a time, but for all your other needs, if following the 60°s, you may get by just running it an hour a day.
With big AH/day requirements, solar is pretty useless, other than helping a lead bank last longer, but since that will only cost a couple hundred bucks, and will likely last at least a year, probably better to keep your eyes on the LFP target.
Of course, getting access to shore power solves all electrickery problems, so also focus on getting creative to get that cheaply.