The problem is that the entire PCB is encapsulated on all sides by a half-plastic, half-aluminum (on the CPU side) cover. A series of long steel pegs goes through the PCB and holds the sides together. If I put too much pressure on it, it may never go back on again / may do some serious damage since the pegs go thru the PCB.

An example of when Intel actually paid a little too much attention to its CPU build quality.
I can slowly work away at the paste residue on the outside from the heatsink; I have all the time and isopropyl alcohol in the world for that