OP, it might be helpful to calculate the child's BMI.
https://www.cdc.gov/healthyweight/bmi/calculator.html
If he/she is overweight the goal should be to stabilize weight so they can grow into it. If they are obese the goal should be to slim down.
You seem to discount the child's weight issue yet vilify your DH for being overweight, I'm sure your DC picks up on that too. The child needs and may want help, that you are not giving with your focus on DH and his language. Stop buying sweets and junk. Have cut up veg and cheese sticks for snacks. You need to put the best interests of your kid ahead of your need to control DH.
Join a gym or buy home exercise equipment, a bike or elliptical near the tv would benefit everyone. Encourage DC to do sports, like cross country that can be year round.