The fact that the fence has been repaired twice, and it appears to have been repaired to an unsatisfactory standard, on its own helps your case.
As already mentioned, forget a solicitor because unless the claim is for over ?5,000 legal costs cannot be claimed back and so any letter or action taken by a solicitor would be at the claimants expense.
But as has also been mentioned, there is no longer such a thing as An Act of God. I cannot remember where I saw it, but this act of god defence was challenged, and it was agreed that it was a get out clause for the insurers who could apply this rule to just about any claim they wished.
Your neighbours insurance will however cover third party risk (assuming he has home insurance) and therefore you are entitled to make a claim against his insurance.
You are entitled to park on your own driveway without fear of your vehicle being damaged, and therefore if your neighbours fence caused damage to your car through no fault of your own, then the insurance company have no reasonable argument to avoid paying out.
It is all about what is reasonable. It was reasonable to park your car on your driveway. It was reasonable to assume that the owner of the fence is liable for ensuring it is safe and secure. It is reasonable to assume that you can park on your driveway and your car will remain undamaged.
You may possibly have a product liability claim against the fencer who carried out the repairs to an unsatisfactory standard, but that is not your concern, your claim is against your neighbour and his insurance, and then if his insurers wish to take the matter up with the fencers insurers then that is for them to decide.