I don't have one in mind offhand but the best source of info will be online rather than opinions which you will get. I'd start with the inverter/charger website then a battery maker website for info.
Don't have one now but have had a solar panel. It should have a controller inline to prevent overcharging.
The inverter will let you know when to recharge. It will shut off when the battery voltage gets too low.
Electronics run better with pure sine wave. This is what your house current is. Again the differences between pure & modified wave can best found on the inverter/charger website.
There are 3 basic types of batterys used by RVs. Wet cell, most likely what you have. AGM, I have a single AGM 4D AGM (no maintenance). Gel cell, no maintenance. There are advantages & disadvantages to each type.
Have fun digging. There is plenty to read.