Thanks! With the board turned upside-down, after loosing the clips, the heatsink (+Peltier) fell off... several IPA-soaked Q-tips later, both the CPU and Peltier are clean from dry thermal paste.

Before:



After:



The much-better-looking board is now being rinsed

(yes, I'm aware that I must clean off that paste residue from the CPU pins!)

Thanks again!