They can't do that.
They can refuse to sign the agreement, and they can lie about why, so the result is the same, but objections based on their understanding planning rules, or their opinions of the aesthetics of HTG conversions are not valid grounds. Clearly they don't want you to do the conversion, realise that if it's PD there will be no planning application to which they can object, so they are looking for anything they can find to stop you.
The PWA is not such a thing - all they can do with that is to cost you money and introduce a bit of delay. If they object you'll just have to go down the party wall surveyor path, but don't waste any time on trying to prove to them that it's PD, just get the surveyor(s) in place, open your wallet, and lie back and think of England.
You could try writing to the daughter, asking her to ask her solicitor which provision of the PWA gives them the right to refuse to sign an agreement if they believe the PW work is associated with work which is not PD.
