Oh, so the 10% hitch weight rule of thumb has been transformed into a 12% ruled of thumb by the weight police. Sorry, I must have missed the memo! All this time I'd thought that the 10% rule was a good one. We'd better sound the alarm to all those Europeans who tow with small cars and 5%- 8% hitch weights (they do keep their speed down, but somehow manage not to kill themselves even without sway control hitches).
Yes, an AGM battery could be used inside. Optima is considered a good brand. I didn't suggest it because the OP already had two good batteries, and not everyone wants to junk a perfectly good battery to buy a new one.... maybe even two, since I'm not sure how the AGM and the lead acid battery will do when charging them together.