If Mueller set up the sting, then, why is he involved in the investigation?
Well they are not admitting that any of it was a sting. Their original claim, which they stand by, was that an "Australian diplomat" told someone (Strzok, presumably, when he flew over the day after launching "Crossfire Hurricane") and so they, as good FBI agents, had no alternative but to investigate it.
This was actually the second theory of the "start of the Trump investigation" after the first one got blown out of the water by lots of information showing it started much earlier.
But these timelines have never been drawn up by the Mueller team or the FBI or the DOJ - it's all been done through leaks, mostly to the NY Times.
It's it's proved conclusively that the spys (er, "confidential informants") were sent by the FBI under Mueller's direction then they will probably just make up some other story explaining that they have REALLY been investigating Trump/Russia since 2015, or 2014...and assert that it's all completely legit. Who you gonna believe, me or your lyin' eyes?