I would be saying to DP, and do, that if he doesn't want to eat what is healthier for me and the dcs to make and buy his own food.
My dp prefers white bread. He likes crisps. He likes pies and pastry and cheap shitty food like kievs and processed pasta and sauce. He enjoys a pudding every day or some cake etc. He likes takeaways and would rather buy a sandwich than make one.
I am a good cook and we compromise for the sake of everyone's health. So I will do a fakeaway takeaway. I make my oen chinese curry, pizzas and thai food. I only buy brown bread. If he wants white he goes to the shop. I only do kievs or processed food on a monday night when we have swimming and the dcs have a jacket potato as do I.
We do have a take away once a week which is too often but we make healthier choices from the menu. So rather than a korma me and dd share a chicken tikka with boiled rice etc. And ds (nearly 4) picks at that too.
I am mostly a sahm so I do the.majority of the cooking. Friday is takeaway. Saturday is easy tea depending what we are doing so the dcs might have a shop pizza or oven chips and nuggets or occasionally a ready meal though they don't actually enjoy the ready meals much and it's just as easy to find a spag bol portion from the freezer and give them that.
I suspect that your problem is takeaways and your dp being a bit of a baby about food. Buy and cook what is the best choice for the whole family. Of he wants something different he makes his own.
Learn a few easy takeaway recipes. B and M sell mayflower chicken curry powder. Piece of piss to make, low fat and sugar and as nice as chinese curry. Homemade pizza is easy too. I make my own bases but nothing too bad in shop bought one. Make your own tomato base using passata, lots of veggie and tuna toppings. Thai curry is easier to make than indian curry. And if you google slimming world fakeaway recipes you will get lots of low fat, low sugar recipes for stuff they will probably like.
Definetly walk to and from school. Start with hometime as they can't do anything about it if you turn up without the car.
I allow unlimited access to fruit like apples, oranges, bananas. Berries and stuff are in the fridge and sometimes earmarked for a pudding so they ask if they can have them.
They can also have hummus, soft cheese, crackers and breadsticks. But have to ask if it's around mealtimes.
I have a 13 year old dd and you do have to be careful about making food a non issue. I always use language like healthy, vitamins, good tooth choice, healthy oils etc rather than bad, sugar, treat, calories, fat.
I tell dd that she is strong and healthy and has nice muscle tone rather than slim, thin, skinny etc.
But even if your other dcs are slimmeras a family you need to make better choices generally. And I know it's another fucking thing to think about and it will fall on your shoulders etc but if you could do with losing a bit too you will reap the rewards too.
Your dp can like it or lump it. He eats what you all eat or he makes and buys his own.