@othervoicesotherrooms - No, the court does not believe everything the ex says. They look at the balance of probabilities. You can call that guessing if you want but it is based on the evidence presented in court, both documentary and verbal.
Contemporary documentary evidence (if there is any) will be given more weight than verbal evidence. If a witness gives verbal evidence some of which is contradicted by contemporary documents the judge is likely to be cautious about the rest of their evidence. And, of course, if a witness contradicts themselves that will undermine their evidence.
The ex does not have to prove beyond reasonable doubt that the records existed and that the OP has disposed of them but he does have to have enough evidence to convince the court that, on the balance of probabilities, that is what happened.
The court certainly won't accept that they are all extremely rare records worth thousands of pounds unless he can produce persuasive evidence. The starting point would be that the balance of probabilities is against him leaving really valuable assets with his ex for nearly 6 months as that is not the kind of thing most people would do.
otherwise it's not worth paying to make a claim
Why not? The fee is low for a case in the small claims court and is added to the amount he is claiming from the OP. So, if he goes ahead and wins, it will be the OP that pays for him making the claim, not him.