cross-browser testing 3rd party scripts on websites

While I have found a lot to cross-browser testing, I didn't find something for my issue or at least did not know, how to accomplish, what i want.

I have a small script, which is kind of a plugin for webshops. The script gets integrated and parameterized by the owner of the webshop and I have to test it, to verify the correct integration by checking the data sent to us via devtools->network of the browsers. This script tracks various information, such as basket information, or what kind of page the visitor currently looks at.

As there are a lot of different shops with custom layouts and I dont have any access to the dev environment of the shopowners Im looking for a way to do a test on chrome for example and the test gets automagically reproduced in the other browsers and checks if the data sent to us is equal to the manually done test.

Is there a possibility to accomplish such things? While one test usually only takes a few minutes and only seldom I have to run more than 2-3 tests for one shop in a year. It shouldn't be a ton of effort as it wouldn't be worth it. so I'm looking for a kind of easy solution to this, which would not take more than, say, a few days to setup and possibly no maintenance.

Thanks for any information on this.