A lot of this is normal, if irritating, sibling stuff.
Don't use a timer as it will just add to the stress. I'm 50, and if I had to listen to a timer ticking away while I was eating my supper, I'd find it very off-putting.
You need no stress at all about meal times. Give them their dinner (involve younger DC in preparing it, if you think that might work), eat with them, clear away anything that hasn't been eaten without comment.
Behaving well at school and letting it all out at home sounds like masking to me (though could also be neurotypical behaviour - some neurotypical children are just more difficult than others, and have to find ways to rein it in at school).
Look at your older child's behaviour, too. Are they always as nice to their sibling as you think they are? Are they using their 'niceness' as a way to get at their sibling in a more subtle way - in that both children will have worked out whom you're likely to believe and of whom you are more likely to think well?
Whatever you do, don't brand your younger child "sneaky" or "deceitful", as that is what they will become if you tell them often enough that this is what they are. Especially not over something very trivial. With the pizza crusts, for example, if the alternatives are either eating something you don't want to eat, or sitting at the table for what will feel to a six-year-old like an eternity, I'd be slipping them into the bin, too.
If you don't buy sweets, nobody can "sneak" them. If you do want to buy them, give them to the DC and say they can eat them whenever they like, but when they have gone, that's that.
Spend time with them both individually. Even if just 10 or 15 minutes.