Photos library & Time Machine (Photos excluded) on same external HDD

If a disk is used for Time Machine backups, the disk is protected by special permissions. Because of this, Photos will not be able to work with the library on a time Machine volume - you cannot enable the "Ignore Ownership on this Volume" flag. You need a separate disk for the Photos library, at least a separate partition of the external drive. See the footnotes in this Apple Support document: https://support.apple.com/en-us/HT201517

1. You can't move your library to a disk that's used for Time Machine backups.

2. If the volume isn't formatted APFS or Mac OS Extended (Journaled), or has been used for Time Machine backups but hasn't been erased, this checkbox will either not be present, or won't be selectable after unlocking. Erase and reformat the drive for this option to be available.

But a second drive would be preferable to a separate partition, because otherwise you will not be able to create a backup of your Photos Library.

That’s not the case for me. My time machine backup is to a completely different external drive.

My time machine backup is on an external hard drive, the Photos library is on the internal MacBook drive.