I totally hear your concern about weight in kids. My difficult child is 17, 6'3 and not sure his exact weight, but he is uber slim. My easy child is 11, she is just now "developing" and heading into puberty. She has had weight issues since about age 6-7, in spite of being tiny as a younger child and being active in hockey since age 6, soccer, basketball etc. I know with her its genetic, both my and her fathers side have weight issues. difficult child's fathers side is super slim and he obviously has their genetics dominating. I swear he is lazy, no activity, eats horrid food and lots of it, yet is a rake. While my easy child does enjoy snacks and bad for you type treats, she also eats healthy balanced meals in ways her brother doesn't.
I think the approach I take now is the approach I'll stick with even as my easy child ages and grows into a adult. I tend to have to find a way to walk the fine line between helping her make good choices in eating and exercise, realize what she's consuming and yet not making her feel something is wrong with her about her weight, not contribute to self esteem problems etc. It can be tricky I know. I tend to try to keep unhealthy foods/treats to a minimum in the house, period. I don't say "no" or mention at all when she shouldn't eat something. I just don't keep it around, except when I feel it is time for a treat etc. I make and serve her meals, even though she is old enough to prepare her own dinner plates, as well as get her own breakfast and lunch. I do it to appear a mom who likes to prepare the food for the family type thing. Even though i'd love her to do her own breakfast and lunch etc. I do tend to ask her do you want A, B or C for breakfast?? And make sure they are all good choices. If she says no, I want (insert something not offered) I tend to say something like "I know you like (Insert name, say Pancakes with syrup) and thats a great idea for a treat on Sunday morning. For today though, its a b or c, so whatcha want today?. Thus giving her knowledge she isn't being denied the food she wants, and inserting a option to think about that food as a treat to look forward to. I like that when I prepare her plates, I get to decide her portion control. I keep things she actually likes as snacks in abundance. On a raw carrot/cucumber etc kick? I keep them chopped and cleaned in fridge for her. Same with whatever fruits she is into at the time.
I have taken to incorporating desire to speak with doctor about weight with her "check ups". At your sons age, there might be a way to finangle something into his thought process. Say he's got some kind of medical thing ... asthma, allergies, whatever. Maybe a suggestion next time he brings up the medical issue: So, that seems to be something that you should ask your doctor about again. Why don't I go ahead and book a physical for you and you can speak to the doctor about it following a thorough check up, which really is something we all have to get into a good habit of doing on a regular basis. It might also be a good time to discuss your bet with your dad! the doctor could chart your current weight, help you set realistic goals, give you some good tips/strategies, and even set you up to have a few appointments with a dietician. I can call for you tomorrow if you like".
Otherwise, i have to say it must be hard at that age, newly independent for medical stuff. If he has favorite recipes, maybe you could say "hey, you know that recipe for chicken you just love? I looked up some recipes on the internet and found a new way to make it with adjustments that cut out tons of calories, fat, carbs (whatever) and I'm going to make it for us all to try".