I asked the BI administrator to confirm that these tables were not referenced in the BI cubes, and they weren't. He suggested that these tables used to belong to a cube that was long since deleted. This means that at some point there must have been a program bug that has left the nametab inconsistent with the DDIC.
1, Exclude the tables from the unicode conversion in transaction SPUMG by adjusting the exceptions list.
2, Manually adjust the SAP nametab.
I chose option 2, since this was the cleanest option and would hopefully leave the system in a better state for future updates.
I found SAP note 29159 contained some useful information on a similar subject. The note suggested writing some simple ABAP code to delete these tables from the SAP nametab tables DDNTT and DDNTF.
Whilst this was simple enough, I decided that I didn't need to go as far as writing ABAP. I manually removed the entries at the database level using SQL:
SQL> delete from sapsr3.ddntt where tabname ='<TABLE>';
SQL> delete from sapsr3.ddntf where tabname ='<TABLE>';
Then restarted the system (or you could sync the nametab buffer with "/n$NAM").
This fixed the issue and allowed the unicode conversion to continue.
UPDATE: I've since found that it's possible to list the contents of the Nametab buffer and delete specific tables from the buffer using the function modules DD_SHOW_NAMETAB and DD_NAMETAB_DELETE.