Depends which legal angle you're asking about.
I've seen people say that this is a breach of contract. It may well be, but no one can know without seeing the contract.
The Equality Act 2010 is more straightforward. The booking wasn't cancelled because of Linehan's belief. Someone has already pointed out that someone else who is gender critical was in the show and that did not cause the booking to get cancelled.
No one credibly thinks that protection of a belief extends to any expression or behaviours of that belief. That was pretty clear in Forstater. So the question is whether Linehan's only crime is believing that sex is immutable, or that trans women are men, and trans men are women. Or whether, in fact, his behaviour is such that a service might legitimately be justified in denying him a platform.
There is a litany that Linehan has done that are dispicable. The nonces quote is just one of them. I would expect the venue to have a field day if anyone tried to claim in court that he was just discriminated against because of his protected beliefs.