VDP will not backup when there is less than 5% free space on any datastore associated with VM being backed up. When a snapshot is created by VDP all VMDKs get a snapshot regardless of if they are being backed up. Are you able to expand the datastore or move VMDKs around?
All backup software using VMware API for Data Protection work the same way when it comes to snapshots. Although some other backup software make it easy to disable the check which might be alright if you change the disks you don't want backed up to dependent. Even if this was an option with VDP I would still recommend increasing free space.