You could do this. The cross check is versatile. You would solve a few of the problems with bar end shifters on the drop bars and some Cane Creak or similar road levers for V brakes. This will work with what you have, and it's cheaper simpler set up with less to go wrong than brifters. Which IMHO is what you want in a commuter. I like brifters on road bikes, but for a daily do everything bike I think there are better options.

On the other hand, I can't imaging the Trek frame will have much resale value, so you'll be sitting on that. Why not sell the Trek as a complete and buy a Cross Check complete, or a CC frame and spend the winter scouring the web for parts, and do your own build come spring. (my preference).