The screen you get when you hit F12 is vastly different than what it is in IE10/Win7. In IE10 (on a VM) it says it's in IE10/standards mode and I don't get the nag. In IE11 the developer tools look like this
. (I deleted some html code from the window so it didn't give away any gmail URLs or other personal info but this is basically how it looks)
edit: oh, wait, here we go. Clicking on that computer-looking icon gives me a document type drop-down. It's selected to 10 by default. Changing it to "edge" is apparently the IE11 mode and it works.
I also managed to get the nag to go away by un-checking the option for MS compatibility lists, so apparently MS thinks IE11 needs to use CV for Gmail. Good enough explanation I guess, though I didn't notice anything weird in "edge" mode.