C15 lost a rivet
 

I realize I can probably get one from Brooks. In the meantime, am I better off trying to find a screw that will thread in there, or just pull the whole thing out while waiting? It's my daily rider.

Brooks will send you one for free likely, they sent me one, send them an email. Can't imagine it wouldn't be ok to keep riding if you keep an eye on it. ;)

You could use a Pop Rivet , too.. it wont unscrew ..

A temp bolt should not hurt, best fit you can and use a lock or rattle washer. Mine loosen to the point I have used blue Loctite on all, problem solved.

What do you grab them with to get them out?

This should help:https://www.ridecrivets.com/install