His insurers have no liability towards you at all. You have no contract with them and they are not liable to meet your losses under “tort”,
Any claim you have is against the Third Party himself. His insurers have a liability to “indemnify” him (that is, meet the costs of claims) against certain losses but they will only do so if he complies with the terms of his policy. One of those terms will almost certainly be that he informs them of any incidents in which he was involved.
Your best bet is to let your insurers deal with the matter.