I don't know what size the hole is but would something like these work? I think their usually called binding screws. The local farm store used to have a drawer of them for fixing broken rivets on saddles and leather.
https://www.amazon.com/dp/B071LFKLZ7/ref=sspa_dk_detail_5
https://www.amazon.com/Chicago-Screws-Leather-Repair-Fastener/dp/B06Y199543/ref=pd_sim_201_1