You're basically where I was at when I built my setup. I had a B-W 6400 series that did the same thing, replace the battery when on shore power. Mine had been butchered to provide a hot to the battery as you are discussing. Here is the problem:
Those things are not current technology, they convert 120v to 12v, but they do it badly. They create dirty, unregulated 12v power and they make a lot of heat doing it. This heat causes them to go into thermal override (at the worst times, it seems) and quit working until they cool down. About 15 minutes or so (you'll actually hear the TO switch click when it resets) If you camp in hot weather this can be a big issue.
So, on top of this problem is the fact that the charge line you are sending to your battery is dirty and unregulated, and it's only going to send a couple of amps forward to the battery when under load from lights, chargers, heater (ESPECIALLY heater) and such. This will be barely enough to maintain, let alone charge your battery. Not to mention the stock wiring will be substandard and there will be voltage loss. It will actually harm your battery in the long run. Deep cycle SLA batteries want certain voltages at certain times and today's charger/converters are built to do that. Yours will continue to send 13.6v forward until your battery boils.
Bite the bullet. Build it right and don't wreck batteries over and over.