He really needs to be assessed for ADHD IMO, going by everything you've said.
This is a good start. But actually if you suspect ADHD I'd just go more in on this.
For the rules - make it really clear and break it down more/provide more reminders than you usually would.
Reward charts great. Exercise great. Diet is probably not going to have much effect if it is not being followed on the other days, so I'd probably go with making sure that he's eating enough/little and often rather than worrying overly about it being healthy. My 6yo (diagnosed ADHD) really needs prompting to use the toilet regularly as well as eating because I think he doesn't pay attention to his body sensations so he doesn't know that he needs to go, but it will put him into a rotten antagonistic mood if he does and he is ignoring it. We actually have trying for a poo as part of the bedtime routine which has helped a lot.
You want to rely much more on positive reinforcement than negative. And curiosity I find helps a lot too. The problem is that if he is doing it out of some kind of desire for revenge or because it brings him a sense of power or control, then the more you come down on the behaviour and shame the behaviour, the more he is going to want to do things which bring him that sense of power or control.
I think this will be difficult to do when you have the relationship that you do - when DS2 (the ADHD 6yo) was 3 and his brother DS3 was newborn, he bit his fingers a few times. Not hard, but hard enough to hurt. Because they were both mine (? maybe?) it wasn't too hard for me to remain neutral and approach it with a curiosity angle rather than getting angry at DS2 or punishing him. Once we had talked about it, all I had to do was say something like "I'm worried about your mouth being close to his hands. I'm worried that you might be thinking about biting him." and it helped to dissipate the feeling and maybe also the urge. Sometimes children (especially children with ADHD) do something just because the thought enters their mind and they want to see what will happen. I think that can be true even when it's something they know that adults won't be happy about and deliberately hide from them - it's not purely an impulsive thing, it can be curiosity over a longer period of time. So if you can possibly approach it from an angle of - don't give him a chance to lie about it, just present him with the information that you know, and let him know you're not telling him off/he's not in trouble as such, but you don't like the baby getting hurt. You know he likes the baby and he likes being a great big brother so what's going on, and could you figure out a plan together to deal with it?
OTOH I remember a longer time ago when I was helping my SIL out with my nephew, helping him with his homework which was tricky for them to manage and he seemed to respond better to me being his aunt rather than his parent. But then he did something to hurt DS1 and although it was totally unrelated I was just so furious I could not bring myself to help him any more.
Of course a stepchild relationship is different to both, I'm not saying this to suggest you are like an aunt, just that it can be really hard to do the neutral approach when you're feeling so strongly protective.
Another helpful thing to do BTW is to encourage him to do things which will give him a sense of autonomy and competence, so basically encouraging him in any interests/talents that he has, giving him responsibilities (e.g. chores) etc. Autonomy and competence is most likely what children are looking for when they are engaging in things that seem to be a power/control thing.