Dear Kostya,
First I am not the owner of this organization or responsible of the IT Dept Here, my desktop still have all the Data, but i am travelling all the time and i need them on my mobile.
I would really appreciate if you take a look to this App by ClockworkMod (Helium), It's really fantastic, after i bought the pro Version i was able to sync most of my apps from the old handset to the new one ( Including their Data) without root and it worked like charm - except for AquaMail and two more apps - where backup is not allowed, through this app, i connected the both devices to the computer to enable the app on them, after that i connected them both to my dropbox account, where the Sync process take place.
better than that, although i am pro edition owner of Aqua Mail, i found a free app called Type Mail, and i tried this process (Helium App) between my two handsets and it worked perfectly both ways including attachments.
i think enabling the backup option in Aqua Mail will not break any privacy conditions or policies, it gives the owner of the messages the right to move them to another device if he wanted to, in fact it's a huge advantage for users like my Case.
P.S. my old handset still have the Aqua Mail, and i am still syncing it, just hoping you to decide to enable the backup option, so i can migrate it to my new phone, because really it's two years of business emails and they are very important to me, Just can't loose them.
Just let me know your decision.