What's the diameter of the flat part of the rivet? Mcmaster Carr sells 1/4" flat head rivets either 1/2" or 3/4" long and the head is .500" in diameter and .083 thick, or 5/16" in diameter either 5/8" or 3/4" long with a head diameter of .624 and .104" thick. They are made of C1006 low strength plain steel. Sold in 1 lb boxes (about 35 for the 5/16" by 3/4" rivets to 80 for the 1/4" by 5/8"rivets) for about 6.21 a box.
Don't know how strong (or weak) C1006 steel is.
www.mcmastercarr.com page 3106 of their catalog.