I would remove it back to healthy growth or wood. And bin or burn the damaged/diseased section. The mould underneath the bark sounds like fungi mycelium i.e their 'roots'. Any toad stool or visible fungus?

Thanks for the advice - I think this might take quite a chunk out of the tree as the branch starts quite close to the ground but hopefully it will be worth it to keep the rest of the tree healthy.

There is no toadstool or visible fungus but the wood under the bark is darker than normal and has some white areas on it.