We have been unable to install the Puffin Browser App on our supervised iPads. We have it in Self-Service. Sometimes it looks like it's installing but once the installation is complete it disappears from the ipad. We don't have any problems installing it on managed ipads.
Have you enabled an age-rating restriction? Basically all browsers are 17+, except ones specifically curated for kids. This could also explain why it looks like it installs, then disappears, because it would actually allow install and then blacklist the app, so it's on there...just unusable
Also, why would you want to enable a web proxy on a supervised device? I'm not sure that, in the situations I'm aware of, limited Flash compatibility outweighs totally bypassing an institutional firewall (and in the case of a school, possibly violating CIPA)