The weapons received via the Speed Square game are Aeris' Umbrella and Cid's Flayer. The Mop can be dug up in Bone Village as soon as the area becomes accessible; later, it can be bought in bulk in Junon (Discs 2 and 3).

3000 points is not sufficient, unfortunately, for either weapon. In order to acquire the Umbrella, you must score ≥5000 points on Disc 1. After Disc 1, the Umbrella becomes inaccessible. On Discs 2 and 3, the prize for scoring 5000 or more points is the Flayer. It's worth noting that you receive these weapons only the first time you successfully score 5000 points at these times in the game; subsequent scores above 5000 will yield the same unusable prizes as any game at or above 3000 points.

As has already been mentioned, I'm afraid there's little you can do to improve your chances of scoring 5000 points beyond practicing.