One of the things about insurance is it is a contract, they have a LEGAL obligation to pursue this matter, but if they choose to do nothing, they cannot load your insurance, but may choose to pay you themselves for this small amount, they can only do this with your permission however, otherwise you can force them to pursue other party.
Don't phone other insurers, the man will no doubt "change his mind" about whose fault it is.
The reason why your insurer may pay you is because it is vastly cheaper than litigation, at £300 a letter, then they have THREE months in which to reply, so all very long winded, they say not us, your insurer counterclaims etc etc, 2 years down the line you might get somewhere, but likely with no witnesses(?) it will be settled by each insurer knock for knock, with each paying thier own bills.
The other shortcut, if you have evidence(witness) of fault is to write to him ask him to pay, get repairs done, if he doesn't CC him, as his insurer WILL take notice of a county court summons!!
tabetha