An upgraded converter-charger is not a bad investment.
A
better investment is a battery hydrometer and a
multi-meter with a current current clamp. I have a
this Uni-T B4Q094 UT210E meter but there are many other to choose from under $100. With this meter you can check the charging voltage, the resting voltage and the amount of current being drawn when you THINK everything is turned off !
It is unusual to to find "true" deep cycle 12V batteries. The most common true deep cycle batteries are 6V golf cart batteries.