No. The Elite is a (much) higher priced controller with extra features compared to the one that comes with the console. Both the Elite and the regular controller are sold in normal versions by many retailers. And both the Elite and the regular controller can be ordered in a custom design via the Design Lab.
You mentioned that your Design Lab controller(s) use regular batteries. That means they are regular controllers with a custom design. One feature of the Elite controller is that it has a built-in rechargeable battery (you can’t use AA’s with it).
If that were true, ‘No’ is also an answer. Total Annihilation does not seem like the only option and certainly not an option to use as a lesson. Can’t teach a dead pupil.