Hi! Fellow Alpenlite owner here. A 2007 model. One of the last of the breed.
I am somewhat confused. Do you have a 12V breaker panel like larger boats or are you talking about connecting the battery to the 120V AC breaker panel, in which case don't do it. DC is DC & AC is AC. the two don't mix.
Yes the battery can feed the 12V fuse box directly. I suspect that you are confusing a 12V fuse box with a circuit breaker. Highly unlikely that you have a 12V circuit breaker box. I'd only expect to see one of those in a very high end class A MOHO. Alpenlites are good but they are not that good.
To charge you will need a CONverter/charger. An INverter is a completely different animal that has nothing to do with charging.
You had better check on the voltage & the rate of charge that lithium batterys require. Different battery chemistrys, wet cell, AGM, gel cell require different battery charging settings. Any old converter/charger won't do if you plan to get a long service life out of your battery bank. Lithiums are not cheap like wet cells.
Most of the better converter/chargers are multi stage & are programmed for wet cell or AGMs. Gel cells & I suspect lithiums will require a user programmable charger for the type & size of the the bank that needs maintaining. No, these programmable chargers are not cheap.