Well it's a little complicated.
First, I'll answer your last question: no, a super-score does not hurt your chances. USAFA does this on purpose, and they endorse taking the test multiple times to bring your super score up.
Now, USAFA reports that the middle 50% range for the ACT is 28-33, with a median average of 31. This, unfortunately means that you fall in the bottom 25%. Now, there are about 1050 cadets at the Academy that scored lower than 28, so not everyone gets a 31. But, much like the CFA, they'll want you to shoot to be above average, even if just so. That means you're going to want a 30-31. Now, there are tons of resources out there to bring ACT scores up, and improving by 3-4 points may not be a huge issue for you if you put in the effort.
But, the Instructions to Candidates booklet reports that tests taken in February may not reach the Academy in time. So you don't have much of a chance to take the SAT, as the next date for that test is in March. But, a test taken in February may still reach admissions in time.
Here's my bottom line: Talk to your ALO. They know better than I do. But here's my advice for you here and now: Take the test. If it reaches Admissions in time, you're golden. If it doesn't, it doesn't. But if you don't take the test at all, it won't reach Admissions at all. You have an infintely better chance of getting your score in on time if you take the test than if you don't. Take the test.