Paul, This sounds complicated! However, I do believe you can still use URL parameters in Tableau Public. Sometimes when doing so, you just need to be careful with how you write the URL. Let's use the following viz as an example: Tableau Public
If you get the URL straight from the browser, you can see it is:
If you add a parameter & value to that URL, they get ignored.
Now, if you click on the 'Share' option of the viz and copy the URL, you can see it is:
That's closer to a URL from Tableau Server. However, if you add a parameter & value to it, they still get ignored. You actually get redirected to the first link in this answer.
Not all is lost, though. We just need to adjust this last URL slightly, adding &:showVizHome=no. If we do that, we no longer get redirected and go straight into the viz, which can now receive parameters & values. For example:
The parameter above is called "Limite Puntaje Fraudulento:", just need to replace spaces with %20
That should do the trick!