This isn't really about Audition at all, but your sound device and what it will let you stream in and out. The issue is simply that of getting two separate audio streams into one device and aggregating them as the output - and recording that. And that's the bit that might be complicated, as you'd need to get the feed from your cart software to be available as a source to record in Audition. That is generally where Virtual Audio Cable comes in, but it can be a bit of a sod to set up. Ideally you need to find somebody who's actually managed this already, or ask the SoundByte team - they will undoubtedly have come across people wanting to do this before.
If you want to try it with Virtual Audio Cable, what you're trying to achieve is to get the output from SoundByte to stream directly into an input channel you can select on Audition. Then you record in Multitrack, and have the SoundByte track recorded separately. At that point, a mix-down will be all you need.