ISO web wrapped app(s) for Google Mail and Calendar that have push notifications but don't download any data

Question

Does anyone know which apps are web wrapped for allowing IMAPing to mail & calendar have push notifications but don't download data to the device?
Basically, our enterprise would like to allow users to use personal iOS devices to access mail, calendar and contacts (we're on Google Apps).  However, our strict security requirements are such that we need to make sure the app isn't being "overly helpful" and saving the accessed data in a cache or pulling it down to the device in any way. 
My quick test for checking if apps don't pull down data to reside on the device is to put the device in Airplane Mode.  If you can see your email or calendar
entries with an app when it's in airplane mode, then it's pulling the data down or caching it in some way. 
I've tested the Google Gmail app and the Google Apps Browser from G-Whizz and both fail this test since they cache the last mail and/or calendar entries accessed.
The only legit way for users to access our enterprise mail and calendar is through the browser only.  However, they can't get push notifications or alerts for new emails or from their calendar.  The concern and likely outcome is that users will just set up an account on the native Mail app using IMAP, either knowing and disregarding that it's a security violation or not really understanding that they're committing a security violation.
I gotta believe there's an app for that. 
Any help be greatly appreciated!

Answer 1

The living issue is complex in my case. Legally speaking, I have no personal residence globally. I just live wherever my father goes, which is everywhere. I'm an expat and it seems that these sites aren't really expat-friendly. They seem to be built for localization and assume everyone stays in the same place.
I don't know what I'd be considered. I'm a US passport holder and live in China, but I only speak some Chinese and I can't read it (it using non-latin characters is just too much for my mind's memory to handle). Besides that, I'm not interested in popular Chinese apps. They tend to just be annoying and kooky and generic and put together badly, often full adware and spyware (speaking for most Chinese software, in fact). In fact, a lot of apps I want are illegal in China (YouTube, Facebook, etc) and I use a VPN to use the services. I assume I wouldn't find these apps on the Chinese store. In fact, last I checked, there wasn't many English apps on the Chinese app store. This is probably due to their strict moderation or something.
I've been going to international schools since I got here and college would be in English too. I can speak some Chinese, but not read it.
Well, I guess the answer is that I can't use the App Store in English. So I guess I'll just sign out everything on my phone involving my Apple account and just start using third-party tools to do my business.