Yo Gaijin ~
Think Japanese. Recycle the space under discussion and make it serve 2 purposes.
Think of the Slides used in about all Hotel Rooms in the TV Armoire. The Doors open, but then disappear back into the Cabinet on 2 side Slides. This type of TV-hiding Cabinet also can be seen at lots of Furniture Stores.
Imagine mounting the desired Storage Bin on 2 of these Slides on one side; one above the other. You would pull the entire Bin toward you and then, using a front-mounted Hinge, swing the Bin to one side, out of the way. Now, the 'recycled' space is Bin-free and your Heads have clearance in that space. This idea would allow the Bin to be square, even at the rear.
It might also be possible to use only front-mounted Hinges w/o Slides. However, the rear of a custom-made Bin would have to angle in so that the arc it cuts at the rear doesn't interfere with an adjacent Bin.
For support during travel, I would come up with a lil block of Wood or 'L' Bracket. It would support the non-hinged side of the Bin so as to not try to support a loaded Bin from only the Hinged side. If this lil Wood Block had a slight front incline to it - like a Loading Ramp - the Bin would creep up onto it to rest supported.