I bred 4 villagers.

Waited for them to grow up.

Placed 4 Fletching tables.

The 4 villagers linked to the fletching tables.

I sold one instance (32) sticks to each Fletcher.

I saved and quit to make sure their professions were locked in.

I reloaded the world.

I broke the 4 Fletching Tables.

I moved the Fletchers from their breeding area to the area where I wanted them using boats and Nether Portals.

Once I got them in the area where I wanted them, I placed beds (which all 4 Fletchers claimed) and the 4 Fletching Tables.

None of the Fletchers would claim the Fletching Tables.

I saved, quit, and reloaded to no avail.

I turned them into zombies and cured them, hoping the game would rewrite their tags correctly, but that also didn't work.

I tried breaking all of the beds and fletching tables and replacing them, but that didn't work either.