"What do you recommend as "proper deep cycle battery"?
Again, I'll recommend two 6v "golf cart" batteries wired in series
if they will fit in your compartment and
if you're ok rewiring them to series. Then any charger intended for 12v lead acid batteries will charge them fine. You should be able to get them under $100 each brand new.
They will take more abuse/have a longer life and more usable Ah than any 12v "marine / deep cycle / RV" battery.
True deep cycle batteries are going to be even bigger and heavier than these but the golf cart size are in kind of a sweet spot for RV use. Cheap, tough, and good performance. You should get a number of years' continuous use out of them depending on how hard you cycle them.
If you're going to buy new batteries before you have ironed out the bugs / learned your system, then again I wouldn't spend a bunch of dough. Another option is to just get some Walmart marine battery direct replacements to use while you figure out what's going on. They should work fine just won't have the lifetime or capacity of the GC batteries. I think they're warranteed for a year, so if you kill 'em in a month....
Bad news for lead acid batteries is being drained down too low and too often, also being overcharged for too long and boiling off all the water - older 'dumb' battery chargers will do this.