Let's see if this helps you visualize what I'm imagining. The drawer has to slide out x-amount so it lines up the stealth door in the drawer base with the chance to open it up at just the right spot to get access to the pocket. There's nothing saying you need a bottom below the drawers. You can do two long drawers, same length each, a longer than the other drwaer, yes, in this case having drawer access from either side, or just one long drawer. Me, I'd go 60-40 or 70-30 with the bigger number favored to the passenger side with the shorter drawer for the driver side for quick emergency access stuff. You only need something a little bit taller than what you had there and you can retain the headroom for the dog.
Don't make me drive down there when we're done @ Disneyland in November. Heeheehee