Nominate someone qualified and when rejected nominate his dog, his cat, his canary until the Senate gets a clue and then try another qualified candidate. If rejected try his horse, goat, two sheep.
He could make a recess appointment. He could leave the seat vacant.
Never give up. Never ever ever give up.
There is an argument which I will mention. If the President failed to make a timely nomination, since the text of the Constitution says 'he shall nominate', then, in theory, he could be impeached for failure to perform a constitutional requirement. Of course, that will not happen. If they hated him enough to want to remove him, they would use a more powerful argument.