Today I checked at the local Lowes and Home Depot for Brass Flathead Rivets. The only rivets I was able to find are steel pop rivets. While these would likely work, they are not asthetically pleasing to my desire for authentic feel. I have not found an Ace Hardware yet, so that's last on my list of local places to check. I was honestly hoping this whole issue would cost less then 5$. Seems not.
Level42: I still have the old plate. The way I removed my plate was to squat down, with the door upside down balenced between my calves (I make a great sawhorse too!). I took a small paneling/finish nail, and placed it on the back side of the rivet in the middle. Then I wacked the nail as hard as I could to knock the pin out forward through the front of the door. Most of them free'd themselves with one wack, others took two. There was minimal bending to the original plate, and no no damage done to the door either. I also remember taking some needle nose plyers and trying to sqeeze the backside of each rivet to make them break off, but I don't remember that working well. In the end, the finishing nail in the backside did the job.
I'll gather these measurements and send them off to that rivet supplier. He told me via email there's a part you buy which is $18, which you use with a hammer to unfold the back side of the rivet to get it to expand. These parts are specific to the rivet's diameter.. so yeah, I'll be waiting to get the rivet sample before placing that order. I'm fairly certain a 1/4 rivet is what we want. More info soon.
-csa