Hmmm...I could see the 4200 pack getting hot I suppose, but that 6000mah 60C pack should be more than up to the task. Mind you, the packs DO get warm when they are run. As much as 100F or even a little higher.
You say "...then the next run having the battery cut out after only a few minutes." So, you've put a fresh Lipo in the car, and it only runs for a couple minutes, then stops? No power to the motor? How 'bout power to the steering servo? Still active? Or is the whole car shut down?
You know what.....here's an idea: If the whole car shuts down, and you can get it going again by toggling the power switch, you could have a bad power switch. I've got one going bad on my SCT. Runs fine for a while, then all of a sudden it just stops. No power to anything. Toggle the power switch, and all's well again. The power switch is a cheapo that I've seen fail before on another car. I replaced it with a
HobbyWing.
Of course, this doesn't address the supposed hot battery issue.... I can't believe that a hot battery would cause the ESC to stop the car. A hot ESC, yes. Hot battery? No. The ESC doesn't care, nor even knows, if the battery is getting hot.