((Quick post, might made some edits later if I get some time.))
Walk into the hole as I cut. If it looks/feels like I've broken through or am close to doing so, try to point the gun to the side (away from where I'm drilling) and cut power to the beam.
If and when the external capacitor runs out of power, switch to linked internal/external mode and hope the internal capacitor has had enough time to recharge. Then keep moving forward, as fast as I can. If there's a barrier at the end, apply mining laser and punch through it. Kinamp punch if regular doesn't work.
Finally, I have a few kegs of HMRC standard with me. As I move, try to crush them open and use them to cause the molten tunnel below me to cool and solidify while also spraying the absurd amounts of flour I got around. Steve said there's little oxygen here, so it shouldn't be able to burn. If I run out of flour, throw sample containers instead. ((See? I told you pie making materials were vitally important. Why? Because:))
Even if molten slag fall from above the layer of flour and other debris such as broken barrels covering the solidified rock will cause a weak layer to form, allowing me or Lars to more easily punch or laser through it later.
And molten slag falling inside the brain chamber should not be as much of a problem if I'm not directly over the brain because the molten rock is going to touch the fungi flesh which should be full of water and thus capable of absorbing some of its heat and turn the molten rock viscous or even cause some of it to solidify, while at the same time sagging due to loss of water and creating a bowl for the slag to accumulate (remember the video with the red hot ball and the watermelon you posted? The ball barely penetrated that thing due to its water.). Besides, the brain should have drilled itself some drainage holes through the rock. Or have a giant chasm under it if it paid attention to its supervillain lair architecture class.
((This plan is too crazy to fail! Well, not as crazy as tase the brain with the only rainbow cannon mode I know very little about, but still.
Now all I have to hope is that I don't get unlucky and drill directly over the brain...))