Just for fun, I googled to find the dealer and the reviews published about the dealer. The reviews suggest that the dealer is highly regarded.
The bike sounds like one of the so-called auction bikes, sold as a used bike by a non-Triumph dealer, with virtually no miles on the clock. This should not be an issue because there is a factory warranty.
The problem here may be the usual resentment of an authorized dealer to troubleshoot an electrical problem on a bike where they made no money to begin with.
Warranty labor reimbursement from the factory for trouble shooting is very low. Flat rate repair for a simple problem is also reimbursed at a low rate.
So you may be the victim of a dealership that already knows that you will travel to save on the original buy, and they may just be leaving your bike on the back burner until they don't have enough other work to keep people busy. I'd make sure that Triumph's USA importer is helping them stay focused .
FWIW, once the repair has been correctly diagnosed and repaired, the bike should be a joy.
It's only two weeks to the end of the month. That's not too much longer to wait. I don't think their diagnosis makes any sense. But if it ends well, you can chalk this experience up to tuition.
Best of luck. Please let us know how it turns out.