If you find yourself stuck in a continuous loop of logging in there are several things you can try.

1. Update the Redgate Client Service to the latest version.

2. Open the application and when the browser launches set Edge as the default.

If that doesn't work try copying the web address from the browser pop-up and pasting it into an incognito tab (Ctrl+Shift+N) inside Edge. 

3. Ensure that the licensing endpoints in this article are allowed, Licensing & update URLs to add to your AllowList in firewall.

Alternatively you can activate using fallback:

• Open SSMS
• In a browser different from the one in which the licensing screen tries to open, navigate to http://localhost:22221/redgate/status.html 
• Next to the tool entry on that page click Activate Using Fallback
• Follow the steps