If you're willing to use entrance paths instead of exit paths, Sec 3.1 of this paper gives a precise definition of the combinatorial version of the entrance path category of a regular CW complex: arxiv.org/abs/1510.01907. For my purposes it was enough to use a poset-enriched category, but it is straightforward to turn this into an $\infty$-category if you want. The fact that the classifying space of this category is homotopy equivalent to the original complex is Prop 3,3. I'm happy to say more in an answer if this is what you're looking for!