Converting a simplicial set into a simplicial complex

Let $\Sigma$ be a~simplicial set with finitely many nondegenerate simplices, homeomorphic to a $d$-sphere.

I would like to construct some ''derived'' simplicial set $\Sigma'$ that is already a simplicial complex, together with a simplicial map $m: \Sigma'\to \Sigma$ that is homotopic to a homeomorphism. Is such construction possible / algorithmic?

My attemps:

I found in Fritsch, Piccinini, Cellular structures in topology that the barycentric subdivision of $\Sigma$ is a regular simplicial set and its geometric realisation is a regular CW complex. This sounds nice, as a regular CW complex can be subdivided into a simplicial complex.

However, I'm not sure how to do it to stay in the realm of simplicial sets: formally, the second barycentric subdivision $Sd^2(\Sigma)$ is not a simplicial complex in general. In this text, page 90ff, I found a claim that $Sd^k(\Sigma)$ is a simplicial complex for some $k$ iff $\Sigma$ satisfies that faces of nondegenerate simplices are always nondegenerate.

I think that I could somehow preprocess degenerate faces and change them to be non-degenerate, and preserve the topology by erecting cones over them, but I'm not sure to which extent this works...

Thanks for possible hints.