Having thought about it a bit, my own personal take on it becomes pretty clear. You filibuster. It's the least unconstitutional option, and as a pragmatic matter this is the route that has the highest probability of turning out well in the end.
Recall that the constitution requires the "advice and consent" of the Senate for SC nominees, but does does not specify exactly how the Senate must arrive at its decision or what proportion of the members must agree. And the constitution explicitly states that each house of congress may choose the rules of its proceedings (subject to certain requirements, none of which apply to judicial nominations).
So it seems that a clear argument can be made that filibustering a SC judge is constitutional. It's then a question of whether Senate rules allow for such. I don't think the rules do, but I'm not a trained parliamentarian so I wouldn't really know for sure. But at worst, the matter comes down to a question of violating arbitrary internal Senate rules vs undermining the constitution. Seems like a pretty easy decision to make.
At least, that's my take. I'd be interested to hear from others.