If the tree's endangering the house it's time for it to come down. Speak to the neighbour though and see what they say. It might not change an inevitable result but even so they'll appreciate the involvement in something that affects them.
Think of it as an opportunity to plant a new tree.
Previous parental home had a giant monkey puzzle tree. Looking at google images I can't find one quite like it, ours was at least as big as the biggest ones I could see but it had the great shape that only seems to show on the juveniles (older ones seem to only have branches at the top). I think it was in a book or a paper or something because we'd get a couple of people a year turn up to look at it.