Difficult to see the grounds of appeal. The judge got the law correct; and the standard to appeal on a matter of fact is astronomically high.
The major sin was that the practice didn't follow its own policy. Following policy is really important for organizations as it shields them from claims like this. Once you go off script it gets very hard to defend a claim when there's even a whiff of discrimination.
I don't see that GC beliefs here got any more privilege than another protected characteristics. I can quite see that a trans-identifying customer who was difficult with front-line staff and whose service was subsequently terminated wouldn't need to find very much evidence of anti-trans bias to make a claim of unlawful discrimination stick.
As the judge wrote (para 117 of the judgement):
Discrimination law has a different burden of proof than most types of litigation. The first stage is for a claimant to prove on the balance of probabilities facts from which a court could conclude, in the absence of an adequate explanation, that the defendant has committed an act of unlawful discrimination.
Note could not should or must. It's a lowish bar.
Then in 118 he goes on:
If the burden of proof shifts, the defendant must show that it did not commit those acts and [or?] that the treatment was not on a prohibited ground
Adhere closely to written policies in all cases, but especially cases where you know a potential claimant has a protected characteristic that you don't like, and especially especially when the claimant is a barrister with recent court experience (even as a claimant) in the law surrounding discrimination - and you'll be ok.
In this case the protected characteristic was belief, but the lesson applies equally to all protected characteristics.