Insulin is mainly triggered by the gut receiving sugars and sending a signal to the pancreas to make insulin. Also having glucose directly inserted into the bloodstream can also trigger the pancreas but not as strongly as the gut.
Assuming you aren’t swallowing your mouthwash, I doubt enough (or any) sugar makes it into your bloodstream through the membranes in your mouth to trigger a pancreatic response.
I use ACT daily for dental flouride and a bit of breath freshener. The alcohol in Listerine destroys your oral biom and can cause worse breath due to bacterial imbalances (fun fact: Listerine started as a surgical antiseptic and at one point sold as a floor cleaner).