We had no choice but to cut all these branches off. The rabbits chewed through the bark and ate the cambium on almost every one. The limbs might have leafed out but growth would have been weak and probably would have died in the first hot, dry spell of summer.
The shrub will come back, and will probably be 3 feet tall and wide by high summer... if we can keep the rabbits away as the new shoots develop. (Time for small-opening wire mesh, and/or repellents.)
We had already planned to cut these dwarf burning bushes back before bud-break, to reclaim the view from the deck out to the yard. We knew that meant cutting most branches back to about 12" below the deck rail so the plant would be able to grow all season without needing another trim. In addition, we would have to cut back some branches even farther so leafy growth will develop even in the shrub's interior.
In this case because the rabbit chewing girdled some limbs, we cut those branches below the gnawed section, then pruned what remained as described above.
The shrub will grow back to deck rail level by mid summer. Check back - we'll post photos.