Prestige classes don’t generally get “paired” with base classes, they don’t “replace” a base class. They are just a different class that you take by multiclassing; the only difference is that prestige classes have prerequisites before you can start taking them.
Some prestige classes explicitly stack with some other, chosen class for the purposes of a specific class feature (and that feature only, barring one exception). This is common for spellcasting. In most cases, the class will spell out exactly what sort of classes it can stack with, but they won’t name a particular class—like alchemist—but rather a sort of class feature, like arcane spellcasting or, well, alchemist bombs. Of course, master chemyst is an exception, naming alchemist.
Unfortunately, a vivisectionist’s “bomb damage” so to speak is always 0. That means adding your master chemyst levels to your alchemist levels to determine your bomb damage still results in 0 bomb damage. There is neither a global rule about prestige classes advancing archetype’s swapped-in features in place of the usual feature, nor any specific rules about vivisectionist master chemysts.
(One could argue that master chemyst names alchemist, and not vivisectionist or something like that, and so your bomb damage is actually equal to that of a regular alchemist with as many levels as you have in master chemyst, i.e. you just treat your alchemist levels as 0. But even if we stipulate to this, it does you no good as you have some hypothetical bomb damage with no ability to actually use bombs.)
That said, there is no real reason for a GM to object to master chemyst stacking with vivisectionist for sneak attack. I certainly would if it were me. I would ask.