What can they offer these Senators?
Cover.
Let them offer a deal in which there is a cooling off period after the first cloture failure by a nominee. If the president brings the nominee back a 2nd time AFTER a "cooling off" period of 6 months, then at that point cloture must require only a majority vote instead of a super-majority.
What it boils down to is this. Rather than give ANY nominee an up or down vote, only those nominees the president INSISTS upon after debate and reflection MUST receive an up or down vote.
Or the "half-filibuster option."
I hear you on that sentiment, but it rubs me the wrong way. The Senate has a Constitutional advice and consent rule for the president's Supreme Court nominees. For lesser judical offices, the Congress can legislate that the President appoint the officers directly. Point being, the Constitution doesn't demand a "hands on" role for the Senate for Circuit or District Court judges. The Senate is sticking its hands into the President's business much farther than it is empowered to.
I don't mind the Senate holding an advice and consent role for Cirucuit and District Court judges, but by golly, I think they have a duty to vote.
Could the Senate refuse to vote up or down on a treaty? Could it refuse to conduct an impeachment trial? The Constitution doesn't, after all, recite a duty to do either of those. It give the Senate the power. The only reasonable read is that where the power impacts other branches, with the power comes a duty to exercise it.