You said you don’t want it to interfere with with the paste sitting on your teeth. Your hoping to get flouride absorption into your teeth from your paste. Your mouthwash is designed to do that if it has flouride. I asked if it didn’t because you think it will interfere with your toothpaste.
By the way the alcohol content in your mouthwash (if it is even it) really isn’t concentrated enough to kill the bacteria in your mouth. If you are rinsing with a rinse that has Chlorhexidine then it can reduce plaque, bacteria and gingivitis.