1964 Dart Wiper Switch working "weird"

The breaker is supposed to protect the wiper circuit from overload should the wiper be frozen to the glass or other fault. I might replace it but not delete it. It really has nothing to do with reverse lamps. Evidently you have a failing wiper switch, internal short, worn contacts, but if the wipers do work in all speeds...
Yes they picked off switched 12 volts from the wiper switch to power the reverse lamps in early models. Saved a foot of wire i guess. You could get that switched 12 volts from any switched 12 volt source. Move it away from wiper switch. Later model wiring diagrams will provide info on where the factory moved reverse lamp power source to. Could be the same fuse but directly from fuse box, a weld splice in the harness, etc...