If he’s impeached, “the Senate shall have the sole power to try all impeachments”.
If they can’t try his impeachment (here because of procedural games), is he in fact impeached?
If I’m indicted by a prosecutor, and the prosecutor doesn’t show up for my trial, isn’t my case dismissed?
The problem here is that this is a political process and not a court of law. Political means that number of votes drives the process. As long as men of dishonor (i.e., 'Rats) control the House by having the majority of votes, the process could go on without end. Sure, it could cost them control after the next election, but when all one thinks about is how much they hate their political opponent, rational thinking is usually not in play.
Your case couldn’t be assessed without the indictment being transmitted to the prosecution so, in essence, there wouldn’t be a case or charge to be acted on or ignored.
It would reside where it originated and no action, one way or another, could be taken...