Well I don't have one but if you wanted to get rid of the grease the best results will be from submersion and ultrasonic or submersion and scrubbing with a correct size bottle brush that you are able to get in there. Again, with the complexity of those wafer switches, you are liable to damage something with anything but the ultrasonic. I can assure you that a damaged wafer switch will piss you off more then doing ten rewiring jobs.
You could also try spraying solvent in there for a while, but its basically a kind of plastic, unless their ceramic wafer switches.. its actually possible to damage something. For instance, I had BNC to Banana adapters get damaged from excessive 50% isopropyl solution (they got warped!). What solvent to use in excess on a 60 year old piece of equipment... I don't know if there is a good answer. If you get that freon stuff mentioned in the thread, warm it up, and spray the wafer with it, maybe that would work instead of a dip. Like forceful spraying with a power air source.
Only suggestion I Have for shorted stuff in that kind of equipment to eliminate variables is to wire the meter up with clips and then adjust the dials, pots and manually actuate any relays to see if any mechanical state change results in a impedance change, and to also reverse polarity on the measurement incase there is a diode you missed or something. And give it a prod to make sure there is not a broken wire that is restring on a place that it should not be causing a short. But usually unless there is a temporary shunt across it, a 6 order of magnitude change in impedance is very bad.
And FYI usually the wires going to those switches are on the exterior, so its not that hard. You might only need to desolder 10% of the wiring on a wafer stack to remove it from the chassis and dip clean it. How do you not mess this up?
1) take pictures
2) open pictures in editor or print them, and put labels on the wires, every wire you de solder should be color labeled on your document. They like to use those spiral candy cane wires in HP, so its easy to get confused from camera picture since its not solid color. And the wafer stack could be represented as a circuit element, its a bunch of radial pieces that stack up, with a pin out. Just draw the wafer slices next to each other on a piece of paper in the correct order in the correct orientation with the wires labeled.
3) if there are same color wires (i.e. someones dodgy repair), put heat shrink bands on them to give them unique ID's, and label it in your document.
4) if you make a nice document, shrink and laminate it, then place it on the bottom of the chassis with tape for next time.
If you remove it, you will find it a hell of a lot easier to lubricate/deoxit/whatever the contacts, not as nice as taking it apart fully, but you still get ALOT of access for only 10% of the work. And it will fit in a small ultrasonic machine.
For undoing the wires, you need to get some picks. And a portion of the wires might be cut so close that if you fail at removing them nicely, you will either need to resolder crap or replace the wire. With my luck I only need to replace.. maybe 1% of the wires I remove, because they were cut too short and it looks mangled and too much solder goes under insulation, for HP gear. If you damage a wire, well cut it off, measure the strand with a micrometer or caliper, count the strands, get the same AWG wire and replace it. You will only have a problem if you refuse to get the correct wire and try to put something too heavy in there or put something too thin that fuses/overheats.
Also if you see a wire that is clearly long enough, sometimes cutting it with flush cutters on the contacts very precisely and carefully, then removing the crap and restripping the wires is better then trying to unhook it, if you feel like you can break the contact of the switch because its being annoying, replace the wire!