It would appear that insurance company of the driver that hit the fence investigated the accident and that their insurance adjuster has determined that their policyholder was not negligent and that the actions of the driver that forced their driver off the road was the one negligent and responsible.

Since the driver's insurance company is not taking liability for the accident and denying the claim then they will not pay for the damages to the fence. If your company feels that this person is responsible they may want to use legal actions to go after that person personally for the costs of repairs and let a court decide liability. A liability lawyer familiar with your state's laws should be able to provide you with more information on this situation and what steps to take at this point.

State laws differ and without knowing the exact specifics of those laws and the situation that you described we cannot say who should be responsible for the fence repairs. A lawyer as suggested above or perhaps your state's insurance regulatory body may be able to give you consumer advice.

It would appear that there is a misconception in regards to no-fault insurance in this case. No fault insurance would not come into play with regards to the accident you described since there does not appear to have been any bodily injuries.

No fault insurance deals with car insurance laws that require the insurance companies of each person in an accident to pay for medical bills and lost wages of their insured, up to a certain amount, regardless of who was at fault. The effect states are trying to obtain by no-fault insurance laws is to eliminate lawsuits in small accidents for injuries sustained in the accident. 

While each state that has no fault laws under most current no-fault laws, motorists may sue for severe injuries and for pain and suffering only if the case meets certain conditions. These conditions, known as a threshold, relate to the severity of injury.

Also the term no-fault in some states is used to refer to personal injury protection (PIP), the portion of a person's auto insurance policy which is basic coverage for the insured's own injuries on a first-party basis, without regard to fault. If the person's insurance company had accepted the claim then it would have gone through the driver's property damage liability coverage.