Check with your city, there is likely to be some code or ordinance that covers this and if the tree is planted on your neighbors property they are going to have responsibility for it.
Insofar as what the insurance companies are claiming and the new legislation, it is possible that removal and or damage would still be covered under grandfather clauses. These trees were planted before the insurance company was even established and did at the time if securing the policy cover such damage, unless your neighbor only recently obtained the policy since the change in state law, in which case he would still be responsible for the costs and damage. Unless it falls then there are other arguments.
Call the city today and call public works department and check on your city's codes and ordinances, get an inspector out, have handy the tree service report, which will also show that you have been responsible for maintaining your trees, if not request a copy for the inspector, let the city cite your neighbor and he can be responsible for arguing with his insurnace company and the city will enforce for you.
Since the trunk is on your neighbors property, wait until the city tags it with a RED RIBBON before bursting into IAAL's song.
Here is an example of one cities code re dangerous trees and property maintenance:
Section 23.9. Landscaping maintenance requirements.
All owners of land shall be responsible for the maintenance of all landscaping. This includes mowing and maintaining abutting rights-of-way, swales, lake and canal banks. Landscaping shall be maintained in a good condition so as to present a healthy, neat and orderly appearance at least equal to the original installation and shall be mowed or trimmed in a manner and at a frequency so as not to detract from the appearance of the general area. Landscaping shall be maintained such that it will not cause property damage and public safety hazards, including removal of living, dead or decaying plant material, removal of low hanging branches below twelve (12) feet above grade and those obstructing street lighting. Landscaping shall be maintained in accordance with the following standards:
(a) Insects, disease, etc.: Landscaping shall be kept free of visible signs of insects and disease and appropriately irrigated and fertilized to enable landscaping to be in a healthy condition.
(b) Mulching: Three (3) inches of clean, weed-free, appropriately sterilized organic mulch shall be maintained over all areas originally mulched at all times until landscaped area matures to one hundred percent (100%) coverage.
(e) Replacement requirements: An owner is responsible to ensure that living material are replaced with like material if such living material or trees die, or are abused.
(f) Removal of root systems: Removal of root systems which show evidence of destroying public or private property is required.
(g) Tree Abuse: Tree abuse is prohibited within the City in accordance with Section 12½ - 16 of the City ofXXXXXX Code of Ordinances.
(h) Tree Pruning:
(1) All owners of land must prune trees in accordance with the National Arborist Association Standards. Any pruning performed without conformance to the National Arborist Association Standards shall be subject to enforcement by the City.
(2) All tree pruners or removers that provide services for a fee within the City of XXXXX shall hold a valid occupational license in either Broward, Palm Beach, or Miami-Dade Counties.