Lilli never gets her brain stolen. Because she has a cold, she can't smell the black pepper from Loboto's peppermill arm so she can't sneeze her brains out. So he just kidnaps her which is why Raz has to get to the Thorny Towers Asylum and Dr. Loboto's lab.
Raz doesn't go to the lake to get to the tower. He goes to the lake at night to meet up with Lilli (if you keep going back to Ford's sanctuary to use the Brain Tumbler to finish up other levels, Ford yells at you that you're supposed to meet Lilli at the beach). It's also about that time that he calls you into his observation area and shows you all the brainless kids and says you have to recover their brains at the Asylum. That's when he asks you if you're ready to take on this new challenge and if you say "no", he slaps you until you say "yes" (which is funny).
It's possible for Raz to see Dogan getting his brain removed while Raz is in his own Brain Tumbler Experiment level after he's seen the brainless Dogan in the real world at the beach/lake. I think the makers of the game didn't care that the timeline could end up out of order because what happens in the mind-levels isn't really reality. It's a mixture of the Coach's mind and Raz's mind so Raz may just be seeing a memory from the Coach's brain.