Just to clarify 1 point: there is a log chain on the mirror database. It has the exact same log chain as the principal. Without it, you wouldn't be able to fail over and bring the mirror online. Also, database mirroring will not allow you to do anything that will break the log chain. If you try to do something that would break the log chain, you will receive an error that you can't do that while the database is mirrored.

Another point: Taking a manual log backup of the log shipping primary does not break the log chain. what causes log shipping to fail is that the automated processes may not copy and restore that log backup causing a gap in the log sequence number on the secondary. You can remedy this by manually restoring (with NORECOVERY) the manual log backup on the secondary or by taking a differential backup that spans the missing gap and restoring it (with NORECOVERY) manually to secondary. After doing this, log shipping processes will be able to resume normal processing.

If you do need to take an out of band backup, it is recommended that you use the COPY_ONLY option as this generates the backup without updating the last LSN backed up.