The deltran (battery tender) is one I'm familiar with --- not a fan. My 2012 (I didn' know the year of yours) had one that failed catastrophically. When I discovered it, capacitors inside had blown out and there was even a small scorched spot on the underside -- I assume the the capacitor overheated and flamed out. No fire, thank God.
I've been told that sick batteries do a number on deltrans.
If you plug the rig in, the charger lights you will be concerned with are located between the grey and fresh tanks underneath. You can see them shining down on the floor of your garage if it is very dark, red and green if I recall. I can't find the manual, but I don't have time to look either.
You can't miss the "battery tender" logos on it...although they may be faded or the labels gone in 10 years.
If the lights are off, check to ensure the circuit breaker on the trailer and the GFCI aren't tripped, and of course that the extension cord has power. Dont' ask how I know that. Likely it is this charger that has ceased to be. If the lights are out - its a safe bet its a gonner.
You can also check the voltage at the battery....if you disconnect from the tow vehicle and plug-in shore power you should see 13.3+ volts on the battery as the charger is doing its thing. If you don't see a voltage change...again, pointing directly to a charger that either isn't connected electronically or is dead.
Since you can charge the battery of the TV --- the battery itself is likely fine, or at least healthy enough.