Sometimes you've got to just let go of things. Hard as it is, the young man is obviously at an age where he has to make his own choices. You know it's healthier to eat as you do...but it has to be his choice to follow you along that path. By the sounds of it, he has gone from living with the family, to living with you (maybe some college in a dorm inbetween?). This is probably his first chance at making choices for himself. He has to learn for his own mistakes I think. When he decides to make a dietary & lifestyle change you can be there to support him, but you can't impose change on him. Let him buy & cook his own food & you do the same. Focus on getting your diet & training right. It sounds like you are actually the one suffering as well as him.
He is being a smart A$$ because he realises you are doing the right thing & he's not. But because it takes some effort & commitment that he's not ready to make he is attempting to change your lifestyle to fit his. Your job right now has to be to look after yourself so that when he is ready to change you can be there & in a condition to actually do something positive for him.
I think the best you can do is to be an example, get as fit & healthy as possible & maybe let him know you are always there for advice & help if he ever feels a need to improve himself. Just don't let yourself slip into his habits. Sort out your diet & eating right now. Get to the gym this week, commit to it, set the time aside. Go buy the food you need to fuel your recovery as soon as you can as well. Much as you call him your "Baby brother", he's actually a young man & so it probably time he bought & cooked his own food. Let's hope as he gets older he'll discover that he needs to make a lifestyle change, but cooking things for him (especially unhealthy things) & changing your lifestyle in an unhealthy way because of his influence is bad for both you & him in the long term, so set some goals, hit that gym & get back on that healthy eating program. Consider that the more in shape you get, the more of a motivation it will be for him to follow suit & start along the path towards health as well.
Sorry I couldn't offer anything more solid than that