This ultimately depends on what 'kind' of category theory you want to do; it's one of those studies that can really get as 'large' as you want it to, and this part of what enables us to 'study everything' with it.

Suppose you just want to use category theory to augment your understanding of groups. You can probably get away with just $ZFC$; yes, ${\bf Group}$ is ultimately a locally small category and not a small one, but you can use various tricks ([Scott's trick](https://ncatlab.org/nlab/show/Scott%27s+trick), treating proper classes as their defining sentences in the language of set theory, looking at only groups with certain properties, etc.) to get everything you need in house with $ZFC$.

Now suppose you want to study category theory using category theory. You want to be able to talk about small, locally small, and large categories, and maybe even a few 'levels above this' in a sense that is clear to the categorical intuition but completely undefined in $ZFC$ (outside of absurd stages of 'viewing $n$-classes as their defining formulae', which just feels weird outside of one or maybe two stages to me). You probably could get away with $ZFC$ plus some number of worldly cardinals (corresponding to whatever $n^{th}$ stage of 'class' you need), and this seems (to me) a likely sweet spot for this study. Your suggestion to use the $\omega+1^{th}$ stage is a good one, since $V_{@_\omega}$ is where we would probably want to do $\infty$-category theory for a good ratio of convenience to consistency strength without fussing too much about things that aren't category theory.

Now, suppose you want to study all different kinds of set theories and the relationships between them using category theory. This would require a background theory that allowed you to construct categories of sets in each set theoretical universe, categories of categories in each universe, and categories whose objects are these categories of various kinds from all universes. This is well-defined since (higher) category theory is ultimately the study of structure, and these things have a structure to them, but the foundation involved (if consistent) would have to be able to reproduce models of all possible set theories and thereby prove their consistency, exceeding all of them in consistency strength.

But we're ultimately missing what is probably the most important point here: most working category theorists feel that [material set theories](https://ncatlab.org/nlab/show/material+set+theory) are the wrong 'type' of foundation for category theory (see what I did there). If they work with 'set theory' as a foundation at all they usually work in [structural set theories](https://ncatlab.org/nlab/show/structural+set+theory), and even this is somewhat unorthodox. The standard answer to the question in the first sentence of your post (with the words set theoretic removed), for most working category theorists, is simply 

>[Type theory](https://ncatlab.org/nlab/show/type+theory).

As someone preferential to material set theories for founding all of mathematics I am sympathetic to questions like this, but I feel it's important to remember that we're an *extreme* minority in the category theory community -- I suspect that if you polled 100 'working' category theorists on what the proper foundation for category theory is, you would get very few responses that even mentioned set theory at all.