I assume that you want to execute your code on the current page you are working on without re-setting authentication and environment variables.
Here are the options I found, unfortunately, I could not find a Firefox plugin that would suit your needs.
Disable visual studio from opening a new window during debugging by choosing project settings → Internet → Check. Do not open the page. Wait for the request from the external application . Then restart Firefox
Just join the existing process through the visual studio: Debugging → Attach to the process → click show processes from all users → look for WebDev.WebServer.exe (w3wp.exe, if it is located in IIS) → attach . From there, just reload the page and Firefox should connect to your application. If you want a faster way to make a macro, demonstrated here
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