Thanks for the suggestion, but even after adding a share as \\hostname\SQLBACKUP\database, I'm still getting this error:
Msg 3153, Level 16, State 2, Line 1
The database is already fully recovered.
Msg 3013, Level 16, State 1, Line 1
RESTORE DATABASE is terminating abnormally.
As soon as I change the script to point to C:\SQL_Backup\Database\ it immediately restores.
The databases are fully functional and aren't restoring any logs, so the error message is confusing.