Starting in November I'm going to begin training specifically to reach that goal. Even though it seems counterintuitive to running, I think I may also try to gain an additional 15-25 pounds, seeing as how I am still extremely scrawny, even at my weight. As long as I get strength/explosiveness to go with the bodyweight, there should be no issue.
Mon: Push ME lifting + Tabata
Wed: 4x400m + Power-Biased MetCon (eg. 9x5 heavy Fran, or interval Murph)
Fri: Pull ME lifting + Tabata
Sat: 1600m + MetCon in the 4-10 minute range.
If you want to train "specifically" to set a 1600m record, you have to actually run. Tabatas are great, but running 3200m a week is not going to do it. Also, trying to gain 15-25 pounds doesn't fit into the specific nature of your goal. Your best bet is to join the school's track team and tell the coach that you want to run middle distance events. Then you'll get coaching and a training program geared to races at that distance. If you intend to break the school record, you'll have to do it in a meet anyway.