Even with a dead battery, the 110V system should work with shore power.
If you don't have the means to throughly check out the breaker/fuse box (connections on the backside, breaker integrity, etc.) or the converter/charger, you may have to take the TT to a dealer.
There should be some more members chiming in on the conversation later today, maybe they will have some other thoughts, I'm at a loss without being on-site.
Depending on when you took delivery of the 2012 TT, it may still be under the two year warranty.
2016 GM 2500HD 6.0L/4.10
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Putnam Class V