Category theory from MK class theory perspective?

In terms of consistency strength, Kelley-Morse set theory does not really count as an "absurdly strong" set theory, and set theorists routinely consider far stronger theories.

The consistency strength of KM, for example, is strictly weaker (if consistent) than ZFC plus an inaccessible cardinal, which is the entryway into the large cardinal hierarchy. Indeed, one should view KM as a proxy for an inaccessible cardinal, as if $\kappa$ is inaccessible, then $(V_\kappa,\in,V_{\kappa+1})$ is a model of KM. In this sense, KM is very weak.

For a truly strong set-theoretic background, one should be adopting some very strong large cardinal assumptions.

But closer to answering your question, it follows that all the uses of Grothendieck universes in category theory, which I think is quite common and which amounts to using inaccessible cardinals, are in the sense I mentioned already using some natural models of KM.

When you consider an uncountable Grothendieck universe and all its subsets, what you have is a model of KM.

Morse-Kelley set theory doesn't seem adequate for all the things one would like to do in category theory. It provides a nice treatment of proper classes, so it can deal with large categories like the category of sets or the category of groups. It can deal with a functor between two such categories, like the forgetful functor from groups to sets or the "free group" functor from sets to groups. But the category of all functors between two large categories (or even from a large category to a small one) is in general a collection of classes, and that's beyond what MK can handle directly.

For a smooth development of things like functor categories, Kan extensions, and related concepts, especially if you want to handle iterations, like the category of functors between two categories of functors between large categories, you'll want infinitely many levels of the cumulative hierarchy of sets beyond your large categories. In other words,you'll want not only classes but also super-classes (i.e., collections of classes), super-duper classes, etc. That sort of set theory is stronger than MK. It's still considerably weaker than an inaccessible cardinal, but the underlying intuition is pretty close to an inaccessible cardinal.

Furthermore, once you've introduced all these super-duper-etc.-classes, your original large categories, like the category of sets, don't look so comprehensive any more; you might want a category of all sets, classes, super-classes, etc. Unless I put some serious constraints on my wishful thinking, I'm likely to end up just where Grothendieck did, in a universe full of inaccessible cardinals (while set theorists are looking down at these pitifully "small large cardinals").