I cant comment on 5/3/1 programming with 100% confidence because I am far from an expert on it, but I do know a thing or two about carries. But I would do this:
Stick with 5/3/1 but only do the essential lifts, no reason to completely change anything, but don't expect to progress that much. So for the next 2-3 cycles expect minimal gains in 1 rep maxes, as this is not your goal until the end of August. It should take you like 20 minutes each day to do the squat, deadlift, overhead press, and bench press correct?
I think your primary "assistance" needs to be the progression I showed you above, but you need to gauge your carry strength. I can help you more with this:
Farmers walks/frame carries will gauge your overall carry strength, but will really tax the grip, traps and back. To a lesser extend you will work your walking coordination (what is the word for it) and posture.
Keg Carries are like a standing deadlift if you bear hug it (this is my preferred form). You will work your lower back, glutes and hamstrings most, followed by middle/upper back and chest.
The bag is similar to the keg, but it can be harder or easier depending on where and how you lift. The bag is much easier off the floor, but much harder to hold to the chest since it is "squishy" and you essentially have to squeeze harder to prevent it from falling. It is also easier to shoulder since it is flexible.
Overhead carries will work your core shoulders and arms. You'll be able to lift the least weight with this obviously but they are the hardest in terms of balance.
I have yet to do a Yoke walk so I cannot comment on those.
The first thing you need to figure out are your "maxes" for carries, which will tell you right off the bat where you stand. Try this:
Take whatever implement you can and carry it a fixed distance for 50'. Increment your weight as appropriate until you cannot make the distance. For example, this weekend I did a keg carry (bear hug style) for 150, 200, and 260lbs before nearly chopping my fingers off when trying to get 310 haha.
Take a fixed weight and carry for maximum time. I think 125lbs/60lbs would be appropriate. You don't even need a track to do this, which is great if the weather sucks.
I recommend: shouldering the bag right away and switching shoulders every so often BEFORE you get too tired to maintain form. Then go to the bear hug form with fingers under the bag if both your shoulders are taxed.
Since you don't need to go super heavy, I think the following carrying priorities should be set up:
really light yoke or just a barbell with 35lb plates on it
you are going to need:
back, core and, shoulder strength-endurance first
lower body endurance second
One mistake people make when carrying is they use their biceps too much. They'll fail fast, which is why it is best to go with the shouldering and bear hug forms.