Hmm, I think you might do well to take this to glitches, since there might be experts there on laptop gaming, which I am not.
I do know about combating overheating in a desktop system, but of course a laptop is quite a different kettle of fish.
Re playing on a fast external dedicated to gaming:
of course every little bit will help, but will removing just the little bit of heat that the hard drive contributes be enough?
Not all externals are quite as fast as internals tend to be either, since externals tend to be more for storage in general. (Plus they're usually a bit more expensive as well).
Sadly I doubt that it will alleviate all or even much of your problem, since the items that tend to build up the most heat would much rather be your CPU and your graphics card. Either or both of those could be sending the "overheat" signal out.
(Although, as I said, that somehow raising the bottom of the laptop to have some ventilation underneath, might also help) ; but I'd rather not give myself out as an expert - I'm hoping someone in glitches could possibly give you some more definite solutions.
EDIT: Heh, you'd probably faint if you saw how big the box for my gaming system is Rowan!

It's huge, and the main reason for it being huge is to prevent overheating; so my system has a lot of ventilation space and fans going for it.
Now just imagine how cleverly they must try and design a gaming laptop to be able to dissapate as much heat as a desktop, and bearing in mind that even some desktops tend to overheat if you don't take care of that aspect...
In any case, good luck, I really hope you manage to find a workable solution for your problem.
