I bought a set of tubular rivet tools.
Link They are set up to put in an air hammer, but can be used without that. They come in different sizes for the rivet that you have. It also had a bit for solid rivets, which I used for resetting a couple rivets in my frame (with the air hammer).
My window regulators were not galvanized that I could tell, I have the set up to zinc plate things, so could do that after tack welding, even with the nylon roller in place. The interior of the door isn't a high rust area, so I really wouldn't worry about that.
If you had access to a lathe, you could produce a couple of solid rivets like Phil mentioned above.
But your setup would work OK, assuming the rivets were a relatively tight fit in the hole in the regulator arm.
