So I was actually a bit annoyed by the solution with the acrylic sheet. First of all, if you're stressing the machine, some of the heat is supposed to exit through the keyboard. Second, the acrylic sheet is slightly visible from the front and the sides.
What I ended up doing instead was to use some spare 3M helicopter tape that I had, with which I built some "feet" to put under the keyboard (basically it's just four layers of the tape on top of each other. This has the added bonus of providing a more stable base of the keyboard, because the tape feels a bit like rubber on the surface.
I'm happy now, I think. No need to disable the internal keyboard, and no need to bring the acrylic sheet everywhere I go.
I hope this is useful to other people too.