Butterfly bush pruning
I pruned my bush in late winter (only cut dried flowers off). It is now spring and my bush is still dead except for one stem I missed.
Normally, butterfly bush is tough and pruning can be done any-old-which-way.. See this article: https://www.gardeningknowhow.com/ornamental/shrubs/butterfly-bush/butterfly-bush-pruning-how-to-prune-a-butterfly-bush.htm
However, if you cut the branches in the winter, and then there was a severe cold just after, that could have been damaging, and would explain why the one branch you missed is still alive. At this point, you can cut back the dead portions https://www.gardeningknowhow.com/plant-problems/environmental/how-to-tell-if-a-plant-is-dead-and-how-to-recover-an-almost-dead-plant.htm
and hope for the best.