All a consulting arborist can do for you is look at the trees from the ground on your side of the property line. Unless
he can get permission to get onto the neighbors property, which if the neighbor thinks it will eventually cost him money, will be hard to get.
Cut the limb up and send the neighbor a certified/registered return receipt letter containing a letter notifying him of the hazard. Than keep a copy of the letter and the receipt in a file. Might come in handy if the tree falls and someone is injured by it. Not much help if the tree falls on your house but would be proof that he ignored a hazardous condition.
I had a friend whose neighbor was always bitching about the spruce wind break that sheltered the neighbors house, said it blocked the view. The wind break had been planted before the farm had been broken up into lots, the way the lots ran the wind break ended up being included on my friends property. He got tried of the neighbors whining about it and had a tree cutting party in Earth Day. We fell the trees drug them down to the one field and trimmed then up, piled the branches up cut the trunks up into logs and stacked them for him to cut into boards and then burned the pile once the stumps had been removed and added to it.
The neighbors was now happy until the fist big snow storm came and he could not get out of his driveway as it had 3' drifts in it. He had to buy a bigger tractor to keep his driveway open while the new wind break he planted on his property grew up to shelter his house and driveway.