If it were mine, I'd go with the 2nd choice but would tidy up those savage looking cuts. If removing a whole branch, you need to cut it off close to the trunk and not leave a stub as those will nearly always die and are then prone to disease. This may have been what enabled to fungus to take hold.
The good news is that the fungus is only growing in the dead heartwood of the tree and the living part of the tree (which is just beneath the bark and known as the cambium) is of no interest to the fungus so it will carry on growing until so much heartwood has rotted that the tree falls. Even that could be decades away as every year the cambium lays a new layer of heartwood (xylem) underneath itself so you could end up with a completely hollow but living tree. I'm sure you've seen ancient oak trees like that?