How about President Obama inviting every member of the Democratic Caucus in the Senate to the White House and getting a public commitment, televised, from each of them:

"I hereby solemnly swear that I will vote for cloture on every bill presented by our caucus whether or not I wholeheartedly endorse its aims.  I reserve the right to vote against the bill on final passage, but under no circumstances will I vote to uphold a filibuster of a bill sponsored by our caucus.  If I break this pledge I fully understand that I will be labeled a Republican swine and will forgo all privileges given to me including committee chairs and assignments.  So help me ..."

Why isn't this doable?