I suggest you check local laws. Tree law is famous for judgements in the hundreds of thousands, so don’t go killing your neighbour’s tree without checking your local laws first.
Where I life, if your tree causes damage to your neighbour’s property then you are responsible for it. Your local laws may be similar, rather than getting yourself a massive financial bill you could instead try to get your neighbour to pay for the damage.
Since they are your neighbour, I’d suggest a conversation with them first about the damage, which they might not know about. Read up on who is liable before the conversation.