The situation may not be as dire as it sounds. I believe this
Q&A helps explain what, exactly, is meant by no plywood in the corners. So long as your "corner" is covered by a full thickness of pool noodle and fabric, you are okay.
The rule was worded (and diagrams drawn) with rectangular robots in mind. You may (and please remember, this isn't the Q&A, so please get a ruling on this before you ship your robot) be able to construct the bumpers in such a way that you have one pool noodle length covering three, four, or more sides of your robot by using a segmented and mitred piece of plywood. As the outer edge of the plywood would form a continuous edge, which is continually covered by a noodle and fabric, when bolted to the robot, this may be considered a continuous, rather than segmented, bumper. It would certainly meet the cross-sectional diagram in the manual.
Alternatively, you could put six (or seven or eight as needed) inch segmented bumpers on six sides of the robot, and demonstrate that when drawing a line perpendicular to the bumper back to the robot frame that you have covered more than 2/3 of the perimeter.
In any case, I sincerely hope that you will find a way to make the robot meet the bumper requirements as I'm looking forward to seeing non-traditional drive trains and robot frames developed to meet this year's challenge.
Jason
EDIT: Check out THIS Q&A... you should be okay!