kitbuga
There are a couple of things you might want to try.
First, with respect to Listerine--it is 30% alcohol, Yup! 60 proof! It also contains "essential oils". Both of these components could be a problem over time. Alcohol tends to dry out rubber components--ask anyone who has been forced to switch to gasoline with ethanol additive in it--the rubber seals fail--rather quickly. So immersion in Listerine may cause problems with rubber components--and if you have and Air II on your inflator hose, those neoprene seals may go bad even quicker. You won't know till you try.
The essential oils--eucalyptol, thymol, etc. are actually responsible for the antibacterial effects. They also penetrate petroleum based materials--you got it--the bladder and the seals. Again, I don't know what will happen--just remember, you're not supposed to use vaseline with a condom (pardon the analogy).
Ok, now for what might work. I wash my dive skin and wetsuits with Woolite and it really gets the smell out. It's not a disinfectant and you would have to rinse a lot to get the suds out.
Another product which should really work is Hibiclens by Zeneca-Astra. It is a surgical soap containing Chlorhexidine Gluconate 4%. It is not a prescription. Its dental counterpart is called Peridex .12% dental rinse--It is a presciption and it contains alcohol. It kills bacteria, viruses, molds, fungi, almost anything. It is my soap of choice for washing between patients. It is also persistent--it gets into the skin/surface and continues to kill--a property called substantivity. I know it works on skin. I don't know about its persistence on artificial surfaces.
IT HAS A BITTER TASTE!!! So rinse it really well. If possible take the inflator hose outlet off and the dump valve outlet off when you are ready to rinse it out. Us a lot of water. Don't lose the little gasket thingy that goes into these outlets. Before removing hoses, though back wash the inflator hose with the solution.
You should need about an ounce per BC and then fill with water to its maximum. Leave it alone for about 10 minutes. Then dumb and rinse.
Also, on contact with things like chlorox, chlorhexidine can make a brown stain which can't be removed. Not that you are going to be using chlorox--although you might rinse your ice chest out with it and if the two mix, you'll get a stain.
Try it. If your BC isn't smelling too funky and it doesn't taste bitter, then you probably have a winner.
Finally, Listerine also comes as Listermint with no alcohol. It's not as good an aneseptic but it tastes better than chlorhexidine.
Good luck,
Laurence Stein DDS