Solicitors talking rubbish anyway there awards not agreements, it is a legal determination by the surveyor(s), not an agreement between the parties. Ultimately you're responsible for the damages, the justifiable costs of the professional advice they have had to resolve the situation.
Your issue seems to be that you don't agree with what is in the award, but that's not how this works, I am really confused about your situation.
If you both have a legally appointed party wall surveyor, their job is clear they need to award and regardless of if you agree with it or not they need to award with what is the correct remedy, then you will have your 14 days to challenge it in court.
Do you both have legally appointed surveyors? as its clear, you have a Dispute unless your argument is somehow that the damage was not caused by notifiable work and thus is a matter of tort claim, regardless though the PWA would be a quicker and could be cheaper remedy meaning you'd only be talking a few hours of each surveyors time