You probably won't like this, but I'd app an app bar at the bottom. If would contain a hamburger menu at the left to navigate mailboxes, buttons for Search, Refresh, and New Message, and an ellipsis at the right to bring up a menu showing the Settings page. I'd move the folder name from its current position in between the hamburger and the action buttons to an overlay of the status bar (like the MSN News app does) since I don't (or at least shouldn't) have to worry about those icons most of the time.
In other words, I'd accept the hamburger as a navigational control, but I'd do it in a way consistent with the usability advances that Microsoft designed back in 2010: putting controls where users can reach them and hiding information that isn't necessary all of the time. As a user, I don't think that I'd be confused that the hamburger and buttons moved around from device to device. The key to an adaptive design is to design for the DEVICE rather than the WIDTH of the device. Putting controls at the bottom of a phone makes sense because of how we hold the device -- that doesn't make special sense for a narrow window on a laptop or tablet screen.
I'd also move the Action Center notification indicator rightward until its just left of the battery. And I'd make it work like the battery icon does when there's text in the status bar. If there are no notifications, it would be hidden; when there are notifications, it would show, just as a low battery icon does.
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