How can I copy email from my iPhone when it's offline?

I really messed up. My main email account is a Gmail instance operated by my University as a benefit for alumni. In the process of cleaning out the 20,000 mail message "Archived" into the "All Mail" folder by my various iOS devices I ended up deleting the mail from my "Sent Mail" mailbox and all the old mail I still had in my INBOX. I've already contacted my University email team asking if they can restore deleted email but I'm not holding my breath. What I have now is an iPhone connected to this same Gmail account that has a cached copy of both the INBOX and the "Sent Mail" from before I messed up. So I put it in AirPlane mode and launched Mail. Sure enough the messages are there and not just the headers, every message I clicked on had the full text, but probably not the attachments. As soon as I take this phone out of Airplane mode and connect to the server things will sync and the messages will go poof! I watched it happen on my iPad. So I'm trying to copy the 200-300 messages in these two mailboxes to a safe location so that I can then reconnect to the server and then restore (re-copy) them to their rightful home. The problem is that with the phone in Airplane mode it'll let me edit the mailbox, select all the messages and then move them to another email account/mailbox, but nothing actually happens because the phone is in Airplane mode (offline). I've tried turning off mail for the problem account and then turning off airplane mode to allow my other email accounts back online thinking they may sync the offline move. That doesn't happen and when I do that iOS hides the mailboxes for the problematic and now disabled mail account. So I can't move the messages in that mode, since I can't see them. Is there some way to export the mail or mailboxes to local storage on the iPhone such that I could then get things back online and then re-import them? Is there utility or application that I could use that might help? Keep in mind, the problem gmail account cannot touch the network or communicate with the Gmail server or the messages I'm trying to save go poof.