Do stomach acid leaking into the mouth from acid reflux have a negative effect on antibacterial rinse used to treat inflammation in the gums?

Reflux. Reflux into the mouth should not affect your antibacterial rinse. The acid should kill the bacteria. However, the acid will damage your teeth and irritate your gums. Typically, reflux should not travel all the way into the mouth. Gastric contents that travel back into the mouth is vomiting. Always speak with your physician about your concerns, they can provide the best explanation for you.
Possible. There appears to be a pathogenic link between chronic inflammatory periodontal diseases and the presence of pathological gastroesophageal reflux .(eksp klin gastroenterol. 2010;(8):46-51.)this may counteract some the antibacterial rinse's efficacy. Since you didn't mention which rinse you are using there is also a possibility that the ph change in your saliva may effect the rinse as well.