Found the answer: More detailed post in my previously linked own question here: WDC: Basic HTTP authentication in jQuery AJAX call with JSON-P
There actually is no way to do a JSON-P call without the authentication popup. What I did to make it go away: use a CORS proxy and let it hande the authentication. Unfortunately the proxy needs to be kind of "whitelisted" on the API server. But asking the API provider kindly might make them tell you if they already have a CORS proxy or if not set up one for you
And then, without the popup, you can simply store the username and password in tableau.username and tableau.password and Tableau will handle them from there on.
Is there a reason you want to go down the Web Connector path as opposed to replicating ServiceNow data in real time to a database and then letting Tableau report against that?
Here is a video of the replication technology Perspectium offers.