Well you don't know that their solicitor does say they have a good case, but ultimately, a solicitor has to do what he is instructed to do by the client, however wrong he thinks it may be (unless it is criminal of course!).
You could, if you wanted bring an action for specific performance - ie get the two quotes out of them, but I think the Act provides the answer. I would have thought that your PWA would allow for each side to have a surveyor and would provide that in case of a dispute an independent surveyor be appointed as arbitrator. Even if it does not s10 of the Act provides what you should do linky here
http://www.opsi.gov.uk/acts/acts1996/ukpga_199 60040_en_1#pb3-l1g7. Looks to me like you need to get them to appoint a surveyor and then appoint a third surveyor to determine the award.
Go through the above - and get your surveyor on board with this. Then let them bring court proceedings - as the claimant, they have to prove it - you can defend it on the basis of 1) breach of agreement and 2) breach of statutory requirement. You will still have to pay them something, but I would not have thought it will be as much as they are trying to claim. Thus you need to try and get an assessment of the cost of the work and make them an offer on that basis. It could put them at risk as to costs.