I think that's what I would do. I doubt a repair shop could get a new housing and transfer everything into it
I bet they could, but I don't think he'd like the cost.
This is why cameras with Magnesium bodies cost more, and why pros tend to use them. :-D
I dropped my old Nikon half a dozen times, a few from table-top height, and it had a few notable dents, but never broke the seals, nor affected the performance.
Realistically, just gluing the bit back on and calling it a day is probably the solution. If you don't have the bit, I suggest duct tape.
And I'd point out that all of the best pros I've worked with had some gaffer tape or duct tape somewhere on their equipment. Because it's not the cosmetics of the gear that matters....