Why filibuster is unconstitutional
No, the cleaner constitutional argument is not to attack the filibuster head-on, but to raise serious objection to its entrenchment--that is, to the Senate rule that prevents a new Senate from changing the cloture rule without a two-thirds vote. Senate Rule V provides that the rules of the Senate shall continue from one Congress to the next unless amended by two-thirds of those present and voting.This violates fundamental law as old as Sir William Blackstone, who observed in the mid-18th century that "Acts of Parliament derogatory from the power of subsequent parliaments bind not." Likewise, the Supreme Court has repeatedly held that the legislature does not have the power to bind itself in the future. As the Court stated in Ohio Life Ins. and Trust Co. v. Debolt (1853), for the political process to remain representative and accountable, "every succeeding Legislature possesses the same jurisdiction and power . . . as its predecessors. The latter must have the same power of repeal and modification which the former had of enactment, neither more nor less."
Article I, Section 5, Clause 2 of the U.S. Constitution - Each House may determine the Rules of its Proceedings...
For me the better logical argument is not that the fillibuster in general is unconstitutional, but that the fillibuster of judicial nominees is an abrogation of the Senate's explicit duty under the advice and consent clause, i.e. the Constitution mandates that the Senate "advise" (i.e. report to the Executive) and "consent" (i.e. approve or disapprove). When they fillibuster a judicial nominee they don't do either. These are mandates upon the body no less binding than any other requirement in the Constitution. Therefore, the Senate should not defer to its extra-Constitutional rules to avoid its Constitutionally mandated duties.