The delay seems to be far more a problem for the Democrats than for the Republicans, so let it fester. When the time is right, then consider some action like you suggested.
The problem that I see with your suggestion is that it looks too much like the trial was resolved on a technicality and the Democrats could parade around claiming that they would have prevailed in a "real" trial. A better solution, in my opinion, would be a summary judgment issued by Chief Justice Roberts declaring the case groundless and not worth wasting the Senate's time discussing. Of course, that won't happen. However, the majority leader could push for an early vote insisting that there is no grounds for a trial based on the findings of the House.
I am not saying that McConnel should follow this; only that he could if he wished to. This may or may not be a wise thing to do. He is also free to say that he will wait for a formal presentation from the House. The Senate has complete freedom to make any rules for the trial they wish. The choice is up to McConnel and the Republicans in the Senate. Pelosi can go and pound sand; she has no voice in the matter whatsoever.