I think this all boils down to
Boundaries
Respect
Control
These things are present in any MIL and DIL relationship though sometimes the dynamic isn't always right (I can totally relate to that one) there is a huge power struggle going on here and it's causing conflict and resentment, on both sides.
I think you both need to sit down and have a talk.
On this individual occasion I think that, yes, you behaved unreasonably - she wanted to get the fish and chips as a treat, it was one day, I think you should have sucked it up and let it go, however, I can totally relate to you feeling that your wishes aren't being respected and your concern with the consistency and message this sends to your Son. My PILs got my eldest a selection box on his first Christmas, he was 6 months old, and they try to get chocolate into them at every opportunity, even going against my wishes on occasions where I clearly stated he wasn't allowed any. Now they're older, I think they finally get the hint at times, but ultimately, I saw it as them not respecting my wishes, but I think they saw it as me rejecting them doing something that was them trying to express affection, if that makes sense? And also rejecting or disagreeing or judging them and their parenting. Anyhoo, although we didn't fall out as such, I was very blunt on one occasion with them and I think they took the hint but this was when me eldest was 2, I think if it had got to 9 years, I'd have blown my lid too.
I'd sit down and have a chat, explain your frustrations, whilst acknowledging hers and see if there's a way you can both move forward as positively as you can.
I'm not going to berate your DH as he's in a tricky situation and is very much piggy in the middle in my opinion.
You are going to need to compromise though, your refusing the fish and chips probably felt like rejection and judgement to her, and vice versa to you - you may have to let some things go (easier said than done) but she is going to have to as well.