Actually, the discontinued Firewire version should be able to route individual channels into Audition. The problem may be that, because it's discontinued, the driver won't work with newer operating systems.
However, assuming it'll still work, the first thing to do is go to the Edit/Preferences/Audio Hardware menu. Make sure you have the Alesis drivers and device selected.
Then create a new Multitrack Session. Click on the Input window in each track (it's marked with a right-facing arrow), select mono then (if you're lucky) you'll get a list of all the Multimix inputs. Assign a different input to each track. Once you're set up, I suggest saving your arrangement as a Session Template--it'll save a lot of track selecting next time.
Good luck--hope the discontinued drivers work for you.