I spoke with the graduate secretary in the department of political science, and apparently it would be no problem to delay my public policy comp until August. Doing so would guarantee a ruined summer – spent mostly in libraries reading articles – but may be wiser than trying to cram an adequate amount of preparation into the next three weeks.
Also, writing for the first time in August would mean having one more chance if I failed the exam then. If I write in May and fail, passing in August would be mandatory for remaining in the PhD program. A summer spent preparing to a re-write that I absolutely had to pass would be even worse than one spent preparing for a more reasonable first attempt.