I find that I clean up trees and prune when it works for my schedule. Not, always according to the rules! So, it’s ok to prune your birch now and cut out any dead wood. This will allow light into the tree and encourage other branches to grow. In time, the new growth will cover up this spot. Also, I would go ahead and prune out the branches that are growing sideways.
CAN I PRUNE PARKLAND PILLAR BIRCH WHERE LEAVES AREN’T GROWING ON BRANCHES?
Early spring is the best time to prune trees. But, the problems are always apparent then.