I don't get your scenario but I have to say I never had any issue with using multiple Google accounts on the same browser. I used to have 5+ accounts logged in at once (personal + work + school) and it always worked flawlessly, switching between accounts was always a breeze using the drop down menu on the top right, and not just for Gmail but any Google apps. Microsoft on the other hand gave me endless issues when trying to use just 1 personal account and 1 work account on the same browser.