Lawfag here. Answer is possibly (should be certainly, but nothing in the law works like that especially with crooked bastards in charge).
Technically, this is a "trial" although it does not quite follow typical courtroom format that lawyers and laymen know. The question hasn't come up before so it would be a "question of first impression."
Legal procedures being what they are, this "trial" would likely fall back on other trial processes where this question HAS been answered before. So, the short version is this (and it's NOT a complete answer, the water is thoroughly muddied on this issue):
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Is the lawyer a material witness? If so, lawyer must be recused from representing client so he can be called as a witness. Assuming this to be the case (and we know it's true), Schitt-ass would have to drop off the House Managers prosecution team and step into the box. The other HM's would have to object where they were obligated to do so, in order to protect privileges.
What privileges? IDFK. That's too deep into the weeds for me to answer without getting paid for it - sorry. Answering that question would take moar work and resources than I can frankly devote to it. I don't work for DLA Piper, ya know? (They don't impress but they got scads of clerks and fancy computer systems)
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Is the lawyer not a material witness? If this is the correct answer, then the judge would, in legalese, say something to the effect of, "Fuck off with your subpoena, I got a trial to run and you're just being a dick."