Paul,
I know next to nothing about computer audio protocols, but nonetheless...
-Here's something to easily check: As hinted above, if any other program is loaded and is using ASIO (without the multiclient wrapper) BIAB with ASIO will not sound out ("...ASIO is often a one-trick pony and if you have another music application open that's using ASIO, it can take precedence and stop BIAB from working.) The suggestion was to switch to MME in BIAB.
What is also true is BIAB will not work with MME either, if another program is using ASIO audio, so if that is the case switching to MME will get identical null results. I verified this by using Real Band with ASIO (not multiclient); BIAB would not work with MME, and neither would Windows sound effects, PG's Metronome, etc.
So, be sure you are not running another program with ASIO - check with Task Manager, or verify you're getting windows sound effects, etc.
It seems you have other programs running from time to time, so maybe one is started, minimized, and forgotten. Unless you've got a weird startup group, the suggestion to reboot and go into BIAB first would seem to work also.
Another way of checking the audio output is bypass the mixer and loaded songs. Go to the stylepicker, and audition a real track style within that module. This would seem to bypass muting, track selection, and all that sort of complication.
John E.