The part they need is probably internal inside the engine cases and will require many hours of labor to replace. Triumph will only pay a warranty claim on one repair, which would be your bike. That means the dealer (or the mechanic himself) would have to eat the repair labor on the second bike the part was taken from. While the dealer is an agent for Triumph Motorcycles and should repair your bike under his dealership agreement with Triumph, he didn't design or build the bike and is under no obligation to subsidize Triumph's warranty obligations.
If it was an easy part to get to, he very well might do it. Also keep in mind that the reason the part may be on backorder is that is may have been redesigned or built by a different company so hopefully it won't fail again, and it just hasn't hit the shelves yet.
I know this means nothing to you and you want your bike back now, and I don't blame you. Triumph and your dealer didn't plan on this happening and I am very sure Triumph hates to pay out a single dollar on warranty claims. I could see them stealing this part from another bike if the backorder time becomes unreasonable, a customer goodwill situation so to speak. What is unreasonable?? I don't know, your state may have guidelines for that. If they did steal the part from another bike, it probably wouldn't be at the dealership level, but somewhere up the distribution food chain, maybe even steal one off of the assembly line.