I have not read through every single response but here are my personal recommendations with reasons:
1) Batteries - Get flooded lead cells like the ones Rolls make - Last a lifetime. Learn how to use them, take care of them, you should get 20-40 years with proper care. There is a huge push for new technologies like LiFePO4 and the like ("They are so small!" etc) but these batteries lifespan is very finite, they are not rebuildable, and unless you are severely space or weight limited it makes no sense to do this. Personally, I like batteries that are "buy once and never again".
2) Generator - Responses recommending diesel genset are correct. Look at something like a MultiQuip Whisperwatt, a diesel genset made for continuous duty - ANYTHING that is "standby" duty will not last and will cost you more in the long run when you have to rebuild, replace, or repair it. Get a manual for it, taking care of the engine is like taking care of a compact tractor engine. If you could get by with relatively low power output for cheap, I am wondering if these ~7kW light tower trailers can be converted to generator-only - I suspect that they can - And these are usually rated for continuous duty and people sell them cheap all over the place due to it being configured as only a light tower. These newer diesel generators like the MQ's are more fuel efficient than the old military type/onan gensets you find, I consider fuel efficiency key and better fuel efficiency will mean a cleaner running engine with lower oil contamination rate and less heat.
I would size the diesel for 2x your typical power need; The whisperwatts and similar appear to be most efficient around 1/2 load and will easily beat out the roll-around standby models (i.e. a 12.5kW whisperwatt loaded at 6000W should be notably more efficient than a 6500W honda roll-around loaded at 6000W). Running it at half load also means significantly less thermal stress on the generator side, and since there is an inverse relationship between lifespan and operating temperature, this is a good thing.
I have seen MQ's and similar listed with 20,000 hours on them, original engine, still going strong, so if you get one at around 5000 hours that was well cared for and runs like a top, it should keep running that way if you take care of it.
3) Charging system for the batteries, inverter - Best to do as much research about this as possible and piece it together/architect it yourself. All of the "turn key" systems I have seen which are ready to just be plugged into batteries and a generator with zero knowledge about how it actually works fall into two categories - Great quality and expensive as hell, or poor quality and cheap. You can find good true sine wave inverters secondhand all over the place. Charging lead chemistry batteries is not rocket science.
Good luck. I am in no way an "off grid" expert but have specialized in electronics, including power electronics for past 20yrs all the way from the architecture level down to the semiconductor level so I have lots of opinions