Photos works fine from an external drive. To avoid damaging the Photos Library, an external drive must be formatted in either APFS format or Mac OS Extended (Journaled) format. The drive must be connected directly to the Mac by cable, not networked, clouded, NASed, etc. Additionally, the drive can not have had Time Machine on it since it was formatted. There have been so many problems with using incompatible drives that the newest macOSs won't even allow a Library on a non-Mac formatted drive to open, since there is a chance of damaging the Photos database. See this:
Move your Photos library to save space on your Mac
If this drive is in a an incompatible format, stop running Photos with it immediately! A Photos Library can sit on an incompatible drive, but running it may corrupt the database.
The Time Machine will backup an external drive. When you go to System Setting>Time Machine>Options you can tell Time Machine places to skip, and it automatically includes external drives. You can just remove your drive from that list.