What do you think a libertarian President (fat chance) would do with a Supreme Court appointment if the Senate was controlled by the Democrats, and they simply weren't going to accept anyone who had libertarian views? Keep on appointing people and having them rejected, presumably till his term in office ends? Hold his breath till he turns blue? Reagan appointed Kennedy, who you apparently think is no good, for instance, after Robert Bork was rejected by the Senate. And don't give a BS answer like "A Libertarian President wouldn't have the Democrats in control of the Senate". I know that's the kind of answer you'd like to give.
OK let's say for the sake of argument I'm the Libertarian President with as you say a Senate controlled by Communists (I mean Democrats). I would absolutely do the first thing you suggested, that is keep on appointing Libertarians and having them rejected until my term ends. Why, you ask? Because one of the greatest assets a president has is the bully pulpit. For the entire four year term, I would use that pulpit to exhort the American people to take up the cause of freedom. Every time the Democrats reject my nominees would be an opportunity to drive home the message that I don't expect them (the people) to settle for less from their Supreme Court justices and neither will I.
We Libertarians do things differently from the Republicans and the Democrats, very differently. If the people want more of the same then they should keep voting for Democrats and Republicans. I'm sure a lot of Democrats and Republicans will look at my answer to your question and think, "He's crazy. You can't leave a vacancy on the Supreme Court for 4 years because you're too stubborn to compromise." What they (and you) need to understand is this: there's a time to be stubborn and a time to compromise. The time to compromise is on tactics and strategy, something my party is wrestling with right now heading into the fall elections. For example, do we concentrate solely on low-level non-partisan offices like city councils or do we mix it up and keep going after statewide offices, knowing we can't win them? These are the kinds of questions we Libertarians can agree to disagree on without compromising our principles. There is never a time to compromise on principles.
Which is exactly what I would be doing if I nominate someone I feel will be confirmed by the Democrats knowing full well that this person will not advance the cause of liberty. Besides what's the hurry in putting another statist on the bench? Libertarians everywhere would be expecting me (rightly) to nominate people who share our views. That's the whole point of having power, to use it in furtherance of what you believe in. Nominating someone acceptable to the Democrats is not what I should do. If the people who elected me president in our example wanted me to support Democrat ideas, they wouldn't have voted for me in the first place. They would have voted for a Democrat or maybe a RINO.