I don't think those are going to give you the kind of mixing that would solve temp variations. Even a thin mash is pretty resistive to localized mixing, so you would get really good mixing in the small volume that those beaters will reach, but probably just continuous recirculation of that volume. Pro setups have mash rakes that run through a greater amount of the mash volume. Something like that would be hard to implement in a rectangular cooler.
But I think I'm with Denny, you shouldn't need to mix the mash that much except at the beginning. And with that cooler your temp loss should be minimal over the course of the mash, so losing a bit of heat at the beginning while you manually stir a few times should be ok, no?
I have a 54 qt normal Coleman as my MLT and I stir as adding the grains, once 5-10 min later, and sometimes once more a few minutes after that. Very consistent temps at the end of ths mash.