I just found this utility called BrowserTraySwitch. I’ve never heard of the site, but it looked legit enough for me to give BrowserTraySwitch a try.
BrowserTraySwitch is a system tray utility that launches a dialog for you to select your default browser.
I recommend you configure the icon so that it is always visible in your system tray:
1. Click on the little arrow in your task bar:
2. Click Customize
3. Find BrowserTraySwitch and select Show icon and notifications.
It’s working great so far.