BBB is NOT going to happen. Your own figures show it is more than twice as expensive and for some people they will only use it once.
Official Uno's comes with a USB cable and the Uno can be powered from USB without any problems so no costs there.
100% compatible Uno clones cost:
https://www.sparkfun.com/products/12757
$20 for the made in America one above, or even less for Chinese made (I have some $12 one).
If you want to, you can make your own "Uno" at home with minimal tools. The only "must buy" item is the chip itself available with the boot-loader for $4. No one can make a Beagle Bone at home.
Arduino documentation, support and forums are much more developed: 9 years instead of 3 years for beagle bone or 1+ year for beagle bone black which IIRC was introduced in part to correct "unusual" behavior with the Beagle Bone that was stifling it's uptake. The Uno learning curve is also considerably easier.
The difference is so much that the support available for MULTIPLE people with the Uno may be enough to get it done - for the BBB, it won't be.
As has been pointed out there is essentially no challenge in using a BBB. It can run a full operating system and offers essentially unlimited memory, so it does not help the hidden agenda of this tournament:
To give "makers" a project that challenges their creative programing abilities and to give programmers an introduction to what is possible in the constrained world of microcontrollers.