Be nice.......
GorBlimeyLimey hit the nail on the head. A part is A PART not an assembly. It is a different type of file. Multi-bodies are indead just a modelling tool. I don't see and change occurring in this situation (and hope there never will be, still it would turn into a file translation nightmare for many of us). The only possible LOGICAL use of different mass props in a single part file would be for composite type materials. And that is a stretch - also highly complex if it were truely modelling composites.
If you really HAVE to use a multi body to design several PARTS (though heaven knows why - there are much better and standard ways), you could have a configuration for each "part" and suppress the others, then make an assembly.
If your "parts" have dependencies or are mirrored, etc. you can EDIT part (as opposed to OPEN part) in the context of the assembly, mirror parts, etc. You can even get rid of the in-context relationships that may or may not result from this if you wish, though it sound like you want to keep them.
The software has a lot of tools for doing these types of things ( as with most similsr packages). It is always best to use the methods they INTEND you to use.
Forgive me for assuming, but my trainer/application support engineer/VAR background is making me sense a self-taught user. I recommend getting assistance from your VAR and/or a training class and/or read the manuals/help from cover to cover and look at all the features of the software, noting what they intended purpose is.
Be naughty - save Santa a trip.