You should be getting prompted to enter proxy credentials before Toad even loads.
The first thing I would try is to make sure Toad is not running on your machine. Be sure you don't see any toad.exe process running and if so, kill them. Launch Toad again and look for that prompt. In my testing, I've seen that credential prompt get lost, so when you try this, be sure you're not clicking on any other applications after you launch Toad. Also be sure to check secondary monitors for the prompt.
If that doesn't work, you could try to clean up any app data that might have been created. Rename or delete the following folder if it exists then try again:
C:\Users[User]\AppData\Roaming\Quest Software\Toad for Oracle\14.0
Another possible issue could be that you've got bad credentials stored in your windows credential manager for this proxy. If Toad finds credentials for your proxy server in the windows cred manager, it will try to use those first. I would expect a re-prompt if it fails, but there is a chance it could be failing silently. Clear out any saved proxy credentials for your proxy server and try again.
We have been able to test Toad Subscription successfully under this scenario, so I'm hopeful we can find a solution for you. Let us know how it goes and if it fails again, please tell us what you're seeing right after you try to launch Toad. Do you see Toad show in your task bar?