Its much easier to gain fat than muscle when adding excess calories, even when working out, and its easier to lose muscle than fat when cutting calories. If you overshoot your goal then cut back, you may find yourself with a higher bodyfat than when you started.
I would suggest cleaning up your diet without changing calories too much, then add enough protein and good fat (MCTs) to gain muscle slowly and without overshooting your goal too much. If I understand your schedule, you have about 17 weeks to gain muscle. Realistically, a 5 pound muscle gain would be better than average results, so you only need enough extra calories to 1) counter the additional calorie burn of your workouts and 2) about 100 calories above that to gain the muscle you need. Two protein shakes a day will take care of all of that.
Don’t gain weight faster than about 1-2lbs. per month, or most of it will be fat.