If push came to shove couldn't Mitch just call a recess of the Senate?
Article II, Section 2 of the U.S. Constitution provides that:
The President shall have Power to fill up all Vacancies that may happen during the Recess of the Senate, by granting Commissions which shall expire at the End of their next Session.
Thus, when the Senate is in recess, the President may make a temporary appointment to any office requiring Senate approval, including filling vacancies on the Supreme Court, without the Senate's advice and consent. Such a recess appointee to the Supreme Court holds office only until the end of the next Senate session (always less than two years).