I think I found the problem. I edited my original post, but I'll post it again here:
EDIT: Ok, I did a little bit of online research, and I think I figured out the problem. I'm no electrical engineer (ironically enough, I am a mechanical engineer though) but what I discovered is that somehow the positive and negative sides are FLIPPED on that terminal. So I flipped just that battery around (so the positive side was up against the spring and not the negative side) and it still turned on fine and none of the batteries were getting hot. I would HIGHLY suggest that anyone planning to use the light up feature on this base to check for this and make sure you don't have the same problem. Otherwise, the circuits inside the base could get fried and if that happens the lights will completely stop working forever. Seems very strange that this happened, but it seems to be the case. Great quality control, huh?
Thank you! That's a very helpful post!