I imagine if they'll put that in writing it might be useful. Bits of paper like that often help.
They might just do that, the usual practice is if they find an insurer who should deal they right to them and ask them to do so. The problem is the insurer isn't obligated to deal until judgement has been obtained so you may still have to do this.
I'd day try the insurer again, inform them you are aware they are obligated to settle the claim under the Road Traffic Act 1988. Ask if they require you to obtain judgement. If this doesn't work you can try the MIB and see if that pursuades the insurer to deal, trouble is it takes a while and you may have to make good on your threat anyway.
If you have to issue proceedings write to them enclosing and estimate and asking for reimbursement giving them notice of your intention to issue proceedings pursuant to Section 151 of the Road Traffic Act 1988. You can they issue a claim under money claim online (against the driver) and her insurer will need to settle the judgement.
I appreciate this is an odd one for a first post but the irony is had you been injured and represented by a solicitor your repairs would be settled by now to prevent escalating costs. Problem you have is because its repairs and its less than £5000 no solicitor will be interested as they can't recover costs. That being said, do you have legal cover with your own insurers or through membership of a union, they may help.