Maybe Trump could demonstrate some basis of ownership to justify his standing to demand return of them? Those papers pretty obviously belong to the taxpayers, not an ex-politician. He should demonstrate lawful custodianship or property interest.
I do agree in general that warrants should be contestable in some way before the bell is rang. It looks like they're more or less doing that here, though. Dude was caught red-handed with confidential records after DOJ and the records custodian both tried to get the records in a civilized manner.