Hi William,
I don't know any option to change the backup server errorlog path without restarting the backup server process. The errolog path is stored in the RUNSERVER file which is usually located at $SYBASE/$SYBASE_ASE/install/RUN_<SID>_BS.
Can you please tell us, why you are not able to stop the backup server process? This is usually not an issue because this is not affecting the database availability.
Best regards
Christoph