Just to confirm: there is nothing similar to sp_errorlog for the backupserver.
The main reason for sp_errorlog is that the ASE log can sometimes grow very fast
and over time even fill up a file system, so a method was needed to start a new file
so the older content could be archived or deleted. Backupserver's log really never
sees that kind of growth.
As pointed out by Christoph, rebooting backupserver can usually be done without causing
an outage on a production system, while that isn't true of ASE itself.
-bret