a hardware sound module would work exactly the same as a keyboard which has sounds (not your keyboard which is a controller only, no sounds).

in response to: a) you're describing a couple of different things here, actually. BIAB's backing tracks are either REAL TRACKS or MIDI TRACKS. REAL TRACKS will not trigger a sound module for sounds, and you don't need it to, they're really just recorded loops that BIAB interprets to your chord and timing specifications.

MIDI tracks from BIAB can trigger sounds in either a softsynth (default is coyote which comes with BIAB) or a hardware synth (i.e. sound module) via MIDI output from your computer to the sound module. you'd need to connect your sound module up to the computer using a USB cable or MIDI cables to your soundcard.

b) you CAN connect a sound module up to your Kawai controller either directly or through the computer. if you go thru the computer, you have to deal with getting your latency manageable or you won't be able to play because the delay will be too bad. going direct from the MIDI ports of the Kawai to the sound module will have no delay.

so using a sound module in scenario a), you'd connect both the Kawai and the soundmodule to the comptuer then run audio cables from the sound module to the mixer as well as the outputs you have from your soundcard to the mixer and "mix" them for the desired output.

in scenario b) if you hooked it up that way, you'd have to make sure you have a good computer and especially a good soundcard set up with low latency ASIO drivers with low buffer settings, but it can be done.

you didn't mention what soundcard you have, only that it's external USB connected, so I don't know to tell you if it's capable of low latency set up or not. using a laptop MIGHT be able to get low latency, but laptops are more inherently high DPC latency so it might not work without stuttering, popping and clicking.