Actually, from Dave's tear down photos, 5V at 5A x2, 6.3V x2, -7.5V, 17.5V, -17.5, plus fan supply. As Dave said in his video it would possible to replace the entire power supply with a fully redesigned one with an internal battery. I agree this would be optimal but not cheap. The OP's idea, I believe, is something simple and easy and external so as to not have to hack or replace major components. Four 12V battery packs in series can be done for around $40 plus a charger. Make that $60 for a hack and it is ready to go. Redesign a complete power supply and stuff it inside is going to be a longer process and probably not cheap. Perhaps there is a Kickstarter opportunity for someone here. With enough volume it could be cheaper in the long run than a DIY quick hack. The problem is that it would be mains powered device and need all the relevant certifications. Designing a power supply and battery to stuff inside is not going to be cheaper I would think, if only for a one off or even tens.
Perhaps the idea of a plug in DC-DC converter running from 12V is a good idea. This would circumvent the need for safety certifications and people could use a certified generic power brick for AC use and an external battery for portable use. The problem with this then becomes the loss of AC sync, and hacking the cabinet.
Still, four 12V batteries in series would be the fastest and simplest way at this point. In Dave's tear down I did not see what SMPS chip is being used, but I could be blind or too impatient to watch all of it again.