Jono, you'll find that you may have to disconnect the ground cable from the battery when you're parked at home to save it from going dead because of all the 'little things' in the unit that constantly draw just a little current, various monitors like the propane leak alert and so forth. When you're plugged into 110v, if your voltmeter shows about 13-14 volts coming into the battery, your on board charging system is working. If the battery goes dead pretty quickly without the 110v plugged in, it may be too old and weak to do the job anyway. If it needs replacing, get a 'deep cycle marine style' battery, not a typical car battery, as large as you can afford and will fit your battery tray.
https://www.batteriesplus.com/batter.../bci-group-31m