There is no getting around the legal time required to decide this issue if it is formally asserted.
After pondering some more, I'm not sure you are correct about that aspect.
If, somehow, the Republican Senate votes to have a witness, there will be much more pressure on the White House to allow that than there was in the House. Political pressure, I mean, not legal pressure. It will be harder to have the person just not show up at all. The R senators who put their necks on the chopping block to vote to hear the witness will not appreciate that. And, of course, it would only add weight to the 2nd article of impeachment, it might make it a lot more 'real' to the senators if they were the ones being obstructed rather than the House.
So maybe the witness does show up, but El Prez actually invokes executive privilege. Does that now begin a court battle? Maybe, maybe not. After all, executive privilege is not a blanket that allows a witness to ignore all questions. And it doesn't apply if you are covering up wrongdoing. It's also possible that CJ Roberts could issue an immediate ruling on the validity of the privilege claim. It seems plausible that would be within his powers. It's also possible that the prosecution would tailor their questions to avoid legitimate claims of EP.
Of course, all that presupposes that the Senate would vote to hear one or more witnesses, which I do not expect them to do.
The argument for documents is similar - there will be more political pressure to provide them if the Senate asks than there was when the House asked. And it's possible that Roberts can sign a document subpoena without it being reviewed by the courts (assuming the Senate permits it).
barfo