That’s exactly what I would do. I would also email or snail mail pictures to your insurance agent to show how the trees are before the coming storms.
The problem is, if the tree falls during a storm, they may find a way to call it an ‘act of god’, but if you have proof that it fell a few weeks ago and they’ve had plenty of time to go out and deal with it, then it’s on them.
I don’t know what kind of terms you’re on with the neighbor, the first step could be just walking over and talking to them, but if you ask your agent to send them a letter, you could then walk over and talk to them and start off with “Hey, just a heads up, I noticed those trees and talked to my insurance company to see if they’d cover it if one of them hit my house and they said ‘no’ since it came from your property. They’re going to send you a letter asking you to remove them. Sorry, woud’ve just left it alone if I knew they were going to contact you about it, but once I mentioned it they said they had to move forward with it”.
A little passive aggressive, but it makes the insurance guy look like the jerk and lets them know that they’re on the hook for damage to your house.
FTR, even if you do just go and talk to them, even if you just do nothing, I would still take some pictures. Again, this will help prove that they didn’t fall in ‘this’ storm, but had actually fallen a while back.
PS, tread carefully. Being on bad terms with your neighbors gets really ugly really quickly.