Don't overthink this. In the wild, small trees and shrubs get trodden on and/or eaten by animals all the time, and they don't go extinct because of it. They might not look "perfect" after the damage, but they survive.
A shrub such as redcurrant usually has several "branches" growing from the roots, so damaging one is not going to be fatal. Even if it was completely broken, the bush would probably grow again from the roots.
For that reason it isn't worth making a lot of effort to "save" the damaged part, especially since the damage may be an entry point for disease.
Currants are very easy to propagate from cuttings, so if you have healthy growth you might as well take some as an insurance policy. When the bush has lost its leaves and is dormant in winter, cut off some stems that are about as thick as a pencil and a foot (30 cm) long.
Trim off the top, to leave a piece of wood with about 5 or 6 buds including a bud close to each end. That will probably be about 8 inches (20 cm) long.
Just push the cuttings into the ground for half their length and they will start to grow next spring. They don't need any special treatment or tender loving care, except to make sure you don't plant them upside down!