I am in your shoes. If you want to gain weight, you HAVE to eat more. Preferably, eat as healthy as possible and eat as much protein as possible. I plan on taking the CFA around the same time as you and am following a strength training program.
I strength train 4x a week, doing a push-pull-legs split and a separate upper body day on top of that due to the CFA's upper body emphasis. I do HIIT and cardio as much as I can in my free time to cut calories as well as eating in a slight deficit with high protein. HIIT and cardio prepare me for the 1-mile run as well
Push workouts address push-ups. This is your bench press type workout. I do incline bench press and dumbbell bench press as well. Do some core work.
Pull workouts address pull-ups. This is your deadlift, lat-pulldown machine, dumbbell/barbell rows. Do some core work.
Leg day will address lower body strength and emphasize core strength. Squats, hamstring curls machine, situps/planks/crunches etc.
Upper body day will address accessory movements that should help on the CFA, movements like curls (for pull-ups) and tricep extensions (for push-ups). These are movements that get touched on by your big 3 (squat, bench, deadlift) but could use some more work. Do some core work.
Make sure to do core workouts as often as possible, and do actual running to prepare for the mile run, I do a combination of long runs and short HIIT sprints. Squat strength won't matter if the mile is too slow. Do actual practice basketball throws as often as possible. Periodically run through the CFA to actually measure your progress. I do a lot of cardio to cut weight, but you may want to do less in order to gain weight.
This plan works for me. I have been following it since May and even though I still have room to improve, my numbers have clearly improved over the last two or so months. This plan may not be for you, and there are other great suggestions made in this comment section that are worth trying. If not this, make sure you are doing SOMETHING to prepare. Don't get analysis paralysis over what you want to do, just do something.
The most important part is consistency. If you can't stick to the plan, don't bother. Consistency is a slow process that may seem daunting at times. On dieting, eat protein and eat healthily, and EAT MORE if you want to gain weight. Sleep well, this is summer and you have more time than you would normally. Message me and I can send you a google doc with more specifics about my plan.