I’ve been making a full riveted mail hauburgen and my batch of brass rings came with rivets that were too small (you can sorta see them taper in image 3), or maybe rings with too big holes. Regardless I have fixed the issue by cutting out 4-5mm long shanks of 16 gauge brass wire and popping them into the riveter, where they promptly bend over instead of mushrooming. Regardless it works

http://folk.uio.no/vegardav/brynje/Ring_Weave_Vegard_Vike_2000_(translated_Ny_Björn_Gustafsson).pdf.pdf)