Unable to export photos from Photos

It sounds like your external SSD may not be formatted properly. To avoid damaging the Photos Library an external drive must be formatted in either APFS format (preferred) or in 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 - Apple Support

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.

A Warning: messing around in the Photos Library can corrupt the database. If you want to look around, you should do it with a copy.

Again, "messing around in the Photos Library can corrupt the database. If you want to look around, you should do it with a copy."

Are you using iCloud Photos? Do you Optimize Storage?

You said "it was taking too much space on my iPhone." I don't understand how this is connected with the Mac. We are talking about your Mac, right? Is this Library your System Library? If you close Photos and then option-clicking on the Photos icon, the Library Chooser will tell which Library is the System Library.

To narrow things down, try these steps:

• Restart the computer (of course)
• Re-start in Safe Mode. This bypasses certain potentially disruptive processes, and it often helps. Safe Mode is different for different computers, so see this: 

           Start up your Mac in safe mode - Apple Support 

• Use another library if you have one, or make a small test Library with a few pictures in it and see if the same thing happens. You can create a new Library by closing Photos and then option-clicking on the Photos icon.
• Make a new user and see if the same problem recurs with it.  This is a bit harder, because a new user can't access a different user's stuff. You would need to temporarily move your Library out of your own user's Pictuers folder up to the general "Users" folder. 
• Rebuild your Photos Library--close Photos and re-launch by option-command-clicking the app icon. If you use iCloud Photos, it’s possible that this will cause a re-sync that lasts long enough to make you nervous.

The idea is to find out if the problem is with the Library, with your account, or with caches and login items, and stuff like that. One of these steps may fix the problem, but their failure to work also helps figure out what's happening.

Let us know what happens…

When you say you are "unable to export," how does that manifest itself? An error message, a crash of Photos, a crash of the Mac, or something else? If you get an error message, what is it?