With the help of support we did find out how to prevent students from deleting history with Safari.
It is in the profile under Web Content Filter and then "Enable web content filter" and "Enable Automatic Filtering". This successfully removed the option to delete history in the Safari app, and also removed the ability for the student to do private browsing.
We also need to see if this is possible in Chrome. We have contacted Google support but without much luck so far. Has anyone gotten a profile payload to work with Chrome that might accomplish something similar?
Google's support documentation seems to leave out some information in a couple of places, but it looks like there is a solution using the admin console. Here they say you can manage user profiles for desktop operating systems and omit any iOS or Android. But then here there is a page for managing user profiles on iOS.
Either way, there is a list of policies you can set here. One of which is Clear browser history: "Specifies whether users can clear browser data, including their browsing and download history."