From what i've gathered, their cooling system uses copper pipes (attached to the compressor) which run along our mutual wall. These are plastered in their wall, so i'm guessing if there's any kind of vibration from the compressor, it probably reverberates through those pipes.

That does not sound like its built to code... I would definitely follow through with local inspectors. Other than that, having some rubber or foam insulation at critical cooling system pipes can still reduce vibration noise down the line.

As for the code bit, not sure about your local things... below is some information on restaurant systems and a question at the end.

If its a cooling system for a walk-in refrigeration unit inside there is a setup like this or, alternate photo

From there the heat gets dumped either in to the working area and the hot air gets sucked-up by the hvac or hood systems.(usually only gets done with regard to small refrigeration unit like reach-ins) Or, more often they funnel the health to a box somewhere on the outside. like this At most they make a whirring sound unless something is broken.

Now, there should really never be a dissipation grid "plastered in" to the wall... if its literally plastered in that would prevent any and all heat from dissipating away from the pipes causing the compressors and fans to work overtime and lead to all sorts of problems. (The vibration noise being the least of it all)

Other than that, just one question. Is the noise constant and level, or is it more intermittent with loud "chunk/thunk/clank" noises accompanied by varied "wooshing" type buzzing sounds?

If its this one it might not even be the cooling systems at all... I would bet it would be their washing machine setup. The way some of those work they can send a lot of noise through the water pipes which would come out as a vibration through the wall and floor.