Is there any way to lock down the proxy settings on an iPhone so that they can't be changed by anyone but the person with the password?
Configuration 2 has an option specifically for a HTTP Global Proxy. You will need to get it on the Mac app store (https://itunes.apple.com/us/app/apple-configurator-2/id1037126344?mt=12).
Once you've downloaded that, create a new profile with the HTTP Global Proxy, sign it, and add it to your iDevice.
See this link for more information https://discussions.apple.com/thread/4359121?tstart=0