A BMX caliper such as an Odyssey 1999, Dia-Compe Bulldog, ACS Boa, or any number of others would work. Published reach for the 1999 is 74-95mm and and 86mm for the Bulldog. Some of these brakes are pretty respectable in terms of power/performance, and a lot are garbage. Watch out for weird oversize pivot bolts on some of them.
Other solutions where the reach should be good but that probably won't work very well with a narrow road rim: Odyssey Pitbull and Pitbull 2 roller-cam brakes, or a 990/u-brake adapter and a standard modern BMX u-brake.
Edit: Some of the longest and fanciest of the fancy rando long-reach centerpulls may also work, barely. Compass lists their brake as 65-80mm. However, if it did reach, it would probably be the best brake you could get on there.