Hi. I have had the same issue with my son who is now 12 1/2 yrs old.
My son has been overweight since he was 10months old. Not only very heavy but very tall for his age. My son has Autism, hes on the low end which he is highly intelligent, doesnt have behavioral problems, is very kind and helpful to everyone, he is a great kid but has sensory issues with food. He only eats bland foods with light colors and will not eat any type of meat except beef..no chicken, fish, pork, doesnt eat vegetables, very rarely eats a fruit, his cereals are only the types that lack color such as frosted flakes, rice krispies and mini wheats..thats it.
No other type of fun kid cereal. Anyway, hes in 6th grade, 12yrs old, 208 pounds and 5 foot 9 almost 5'10. He wants to workout and I've had him waiting until he turns 13 so he can go work out at the gym with me. By my side he loves to exercise but seems his weight has never gone down.
I can't wait till he's able to do strength training and work out with weights one day. Burn more calories as he builds lean muscle. For now, its been basic strength training using his own body weight, light weights and stretch bands...it hasn't helped because he consumes more calories than he burns. He's always hungry and eats a lot :-( but at 5'9-5'10 at 12yrs old he's going to continue growing taller and maybe that will benefit in him stretching and getting leaner when the time comes for really exercising with weights.
At least I'm hoping. My tallest cousin is 6'8 and he has a lot of family in over 6ft range. Were Hispanic so that's not common among us. I'm only 5'1.
Nonetheless, I understand your difficulty and concern. With all the gadgets of video games, TV, computers, phones etc..kids rather be on that than exercise. But continue to work on yourself and inspire your child to do it with you at home until they are of age and then go to the gym. Its great that your at least concerned about your child and desire to make a change to improve your child's health..now its time to look at your options or option and take the next step.
Your the example for your child.
Take small steps WITH your child and you will look back and realize how far you have gotten. I wish you the best.