Kolya! Thanks an $\infty$ many times for your question, and for pointing (me) to McClure's paper! :)
You might be interested to take a look at this paper (of mine), where a definition of the notion of a "cat" is presented: the globular (see, e.g., John Baez's An Introduction to n-Categories) multi-simplicial sets satisfying the (multi-simplicial version of the) Kan condition presented by McClure in Definition 5.2 of his paper ( - I think "higher (semi-)Kan complexes" are a good name for them - ) are special kinds of cats, just like Kan complexes are special kinds of $\infty$-categories (in the sense of Boardman-Vogt, Joyal, Lurie, ...).
I did not read McClure's proof yet, but I'm "pretty confident" that what you're asking for should be possible. Also, an analogue of McClure's Theorem 5.3 should be true: it should be possible to canonically build a cat from any given "semi-cat", even if the "horizontal composition" in the semi-cat isn't strict. But as I said, I did not prove this, it's just a guess.
Another question that one could come up with is whether the analogue of Moore's proposition ( - whose formulation should be obvious - ) in this "higher" situation holds.
For several months now I tried to find out whether this definition of a "cat" has already been studied, and if so, why it can't be found in the literature ( - at least I did not find it). If anybody knows of further papers in this direction, for example answering my previous question, please drop a reference in the comments section below ( - if that is allowed).