You can ask. The magistrates will be *very* familiar with people appearing before them for this reason.
Interestingly (I don't get invited to dinner parties very much), what the law says about the identification information is that your name, date of birth and licence number must be provided to the appropriate person. If you hold a UK driving licence, your date of birth is part of your licence number.
The appropriate person for the purposes of identification information is the Fixed Penalty Clerk, whereas the appropriate person for the purpose of paying the fixed penalty is also the Fixed Penalty Clerk. If you provided your name, date of birth and licence number when you paid the fixed penalty, that it would seem that you had satisfied the legal requirement, and any prosecution would be unlawful.
North of the border, Polis Scotland don't purport to issue a separate requirement - you provide the details when you make payment and that is the end of it.