iPad Single App Mode Browser Privacy Configuration

New Contributor III

I have a department that uses iPads for students to check-in to an online portal, and then take a survey (two separate things/URLs). Currently I have a profile for Single App Mode and Content Filter with the two specific websites they need. 

The issue I'm running into is that for the check-in web portal, cookies need to be enabled or the web application doesn't  work/errors out. But the survey needs to have cookies disabled or otherwise have the data cleared from each tab/session in order to be able to take the survey again. Otherwise, it says you have already responded.

I've tested with Safari and Firefox and there doesn't seem to be a way to force Private or Incognito browsing with either of them via Jamf profile. Does anyone have a solution for being able to accomplish the clearing/deleting of site data with either browser? Or some other method altogether? I've considered ditching Single App Mode altogether and just setting Web Clips for each but the Configuration Profile settings for Cookies seem to be universal and either a hard off or on.