I definitely wouldn't choose the Audioengine D1 to power the AKGs. Not sure if the Bravo Audio Ocean is up to the task, or how well either of those amps would work with 600 ohm headphones. Best to do some research on that.
I owned that DT 880. It was very "capable" but sounded utterly boring to me. Too gentle and polite for my tastes. (And this doesn't have a whole lot to do with the frequency response itself, in my opinion, because my favorite headphone -- the Focal Spirit Professional -- is also very neutral. I think it has more to do with the driver design.) Very good build quality and comfort, but the ear pads were so large that they just engulfed the side of my face. lol
I will receive an HD 650 on Monday.
And I haven't heard the K701.
I don't want to recommend things based purely on others' impressions online, but I do know that the HD 650 is one of the most well regarded headphones and that the newer version with the silver drivers apparently isn't as dark-sounding as the original version.
The Philips Fidelio X2 and HD 600 are other worthy contenders in the $200-400 price range.