All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah and that Muhammad is His slave and Messenger.
Your question indicates that you suffer some kind of waswaas (obsessive whispering). You should beware of such whisperings and totally disregard them. Verily, yielding to thinking about them leads to evil consequences.
There is no harm in performing ablution after using this mouthwash and rinsing your mouth during ablution is valid and sufficient regardless of whether the water used in the process changes or not. Hence, you should ward off and totally disregard such obsessive whisperings.
Allaah knows best.