Whilst what the council say is accurate it does not represent the law and the neighbours are entitled to ask you to remove all over hanging branches notwithstanding that the tree was there first.
I understand your concern about both the look of the tree after and the possibility of de stabilising it.
I think I would get ask a tree surgeon to look at the tree under the guise of getting a quotation and ask for an opinion.
If the tree surgeon indicates that I might de stabilise the tree I would go back to the neighbour and tell them . I appreciate they are probably concerned about getting more light in garden etc. but I do not suppose they want a birch tree landing in their lounge anytime soon..
I hope this helps. If there are any further points please reply I will be happy to respond.
Sorry, but if trees which originate in your garden encroach on their garden , whether it be branches or roots, they are entitled to either ask you to trim them or do it