Overlap in LSN's backed up
it looks like like the log records backed up to the nul device isn't being registered on the secondary node of the cluster.


Not to me.  The First Log Backup on the secondary (after the full) starts with the last LSN covered by the backup to 'nul' taken on the primary.

But stepping back to "We had a request to minimize the amount of log backups": why?  And why are you breaking the log chain with the backup to 'nul'?  


  Is this how it is supposed to work?


You are supposed to maintain an unbroken chain of log backups in addition to occasional Full backups.  The log records included in a Full backup will also be present in the next Log backup, as Full backups never break the log chain.  This allows you to loose a Full backup, and restore from a previous Full plus all the log backups.