I would contact the insurance company and get their opinion. And get a quote on taking it down.
Then depending on what the insurance company said, I would approach the HOA (if they are the owner of the tree) and explain to them that it will cost them less to take it down now rather than pay for house damage and tree removal both if it falls down.
But make your case in front of the whole HOA board for 2 reasons. 1 to get your case in front of more than 1 person. 2 to get your case recorded in HOA meeting minutes. I think you’ll get better results from meeting with the whole HOA board rather than a single person.
If HOA balks at taking it down, then try to negotiate 50-50. If that doesn’t work, pay to have it taken down yourself if the HOA allows you to do so.