Advice With A Medical Negligence Claim

[nino]

Hello.
I am the victim of a medical negligence. When I came out of the clinic I approached a specialised legal firm which informed me last week that my Legal Insurance cover will not give them a mandate because they are not members of their panel of legal advisers.
However, the legal firm proposes to "put in place a deferrred private client retainer" so they may proceed on a no-win-no-fees basis and so put forward the claim (which they believe to be clear cut due to one of the doctors having already admitted , in a letter. the oversight).
My concern is this: if the claim is not accepted by the octors insurance we might have to go to court. The legal firm says that at that point they will write to my Legal Insurers and ask them to fund the case, which thing - apparently - cannot be refused. But is this not contradictory ?
If the Legal Insurer can refuse them now becase they are not part of the panel, how can then they accept to take on the cost of the Court cases , legal fees etc etc at a later stage ?
I am of course concerned.
Can anyone advise me ?
Thank you