There is a policy case, and a political case. Strong ones. But the suggestion that somehow nominating and/or confirming him would be contrary to law or violate the Constitution is simply absurd.
A telling admission in the OP: even if Kavanaugh repudiated his prior position it would be meaningless because he could change his mind again. Well, any Supreme Court nominee can change his or her mind at any time, so that's not really much of an answer.
I hope like hell his nomination can be blocked or stalled until after November where, if we're fortunate (and its far from certain), we regain control of the Senate and can either vote down or shut down the nomination (although there's no guarantee of that either, even if there are 51 Democrats in the Senate).
As for the McConnell "rule" -- it isn't a rule, it was a power play by the Republican majority which by virtue of being in the majority could do what it wanted with the nomination. They are still in the majority so they still can do what they wanted (leaving aside the easily made distinctions between a nomination in the year before an off-year election and a nomination in the final year of a president's second term -- although as indicated, those are just fig leafs -- the real point is that the party with the majority controls what happens, or not, to a nomination and short of the voters applying pressure on Congress, there isn't a damn thing anyone can do about it.