Could they vote in secret or Is a roll call vote required for impeachment?
While there are rules, the rules aren't followed. It's a bit like "Rule of Law," being whatever is necessary to get the desired outcome.
The Rules of Order provide a roll call vote "must" be taken if 1/5th of the Senators present agree to have one.
The GOP senators could just absent themselves from the vote. The constitution says conviction takes 2/3rds of those present, but unanimous consent is more than 2/3rds.
I don't think there has even been an impeachment trial that did not conclude with a roll call vote, but just because something is unprecedented doesn't mean it can't be done.
The "whistleblower" complaint doesn't follow the rules (out of ICIG subject matter jurisdiction; not first hand), but that didn't stop ICIG Atkinson from pushing it up the ladder.