This is a known issue with exclusive backups, which is a reason why non-exclusive backups have been implemented. pg_basebackup does that, and using "false" as the third argument of pg_start_backup() would have the same effect. So I would recommend to switch to that.
Is there a plan in place to remove the exclusive backup option from the core in PG 15/16? If we are keeping it then why not make it better?