VEX will not hold up in a battlebot arena unless you restrict it to the point where it wouldn't be fun anyways.
This bot cost me 150 bucks in addition to the reciever and transmitter. It weighs 2 lbs. 7 oz. Good luck to any vex bot to stand up against it. ALSO, most, if not all combat robot leagues require 2.4ghz radio systems on anything with an active weapon. It is a safety precaution, and I don't know enough to judge whether or not it truly is necessary.
Now, you battlebot fans don't get offended when I say that I have been part of both and enjoy FRC much more. Battlebots tends to invite in many people who just wanna break stuff. Most robots are just "lets make a big spinning thing" and then drill holes until it is underweight. Many aren't planned out and do not take much engineering to make (my own included). But FRC, is always a complex challenge with many ways to approach it. It takes careful planning to make a bot work. I am sure Vex robotics competition is much like this (I have never competed in VRC).
However, I do believe a more sport-like game would work. Infact, many FRC games are ready and availible to be scaled down for a vex competition. Of the more recent games, I think Aim high, Overdrive, Breakaway and LogoMotion could all be done on a smaller, simpler scale.