semi-automatic Cross Browser Testing XMLHTTPRequests of a third-party script in a website

Hey fellow overflowers,

Perhaps someone has an idea:

I have a little script, which shall be included in websites, which are not my own. This script sends data to my server on visiting any page of the website. When I now want to test, if the script sends the correct data, I currently do it by using chrome with dev tools open and view the requests manually and then repeat this process with every other browser I need to test. This is kinda tiring and doesn't feel really effective.

What I'd like to do is: Making only one test session manually in one browser and perhaps log the websites visited and extract the relevant requests. Then I'd like to verify everything for this one session and let it be repeated automatically in other browsers and see if I get the same data sent to me.

Is there a way to do this and if yes, how time-consuming would it be to set this up? Is it possible to also get things such as "logging in" or interacting with some buttons on the website automated into this process?

Thanks for any help or sources with further details on it :)