In terms of food LOs do often take in more calories during the earlier hours of the day, breakfast, lunch and snacks, and can show little interest in food later on. You didn't say what time dinner was, I fed mine at 4pm when he was young as any later he would have gone past his appetite for eating and just not bothered with anything. You could try an earlier dinner (although I am also a big fan of family meals and I know earlier is not always possible to fit with the rest of the family - you could feed him dinner earlier perhaps and then invite him to join the family at dinner for the social aspect for a few mins).
I wouldn't skip the afternoon snack at home if it is already a standard part of the routine at day care.
If you are concerned about the range of foods because for instance you cook healthy balanced meals for dinner and he is not eating these, you could try saving a portion over night and serving it for breakfast. it may sound a bit odd but really you can serve anything for breakfast. My own DS took very little protein but I discovered (after advice here) that I could serve an egg at breakfast and he would eat the whole thing. Serve it at lunch and he'd ignore it. I changed breakfast for a good while, making a small one egg omelette daily to ensure he got at least one protein portion at a time of day he accepted it. You might find the same with dinner foods, veg, meat etc.
Another thing you could try is a little appetiser of a small piece of fruit or a few sultanas at the start of the meal. Mine had a phase where he had no interest in eating, just no appetite, but if I got him going with a very few sultanas he'd suddenly be really hungry and eat all his main meal. you can put the same fruit/dried fruit out by your DD so the children are both getting the same, it is her choice to eat before, during or after her main.
WRT discipline. I wouldn't allow a child any age to be getting up and down from his chair or climbing the table. I'm not surprised your older DD finds it unfair. It might be easier for her to understand that she is required to stay at the table a little longer than her younger sibling as she does know he is younger, but for me the rules of the table would be basically the same for both children. You stay at the table to eat, if you are finished you may excuse yourself (and not eat). you may not climb the table or repeatedly get up and down from your chair.
it doesn't need to be a battle ground, he either eats or doesn't but his behaviour needs to have very clear boundaries. Allowing for his age is things like not expecting him to stay at the table for an hour whilst the grown ups have a long 3 course meals followed by coffee - it is not things like allowing him to get up and down or climb the table. I would imagine you can stop this relatively quickly with consistency and a firm tone. Be clear "if you get down from your chair you do not come back, dinner is over" or as soon as he makes the first move towards climbing the table either "sit nicely" or "stay in your chair" or "get off the table" and if necessary remove him before he can get up there. Take him out of the room if needed (stay with him). You might end up with cold dinner a few nights but once you really show that this is not acceptable he will change his behaviour.
Trying to make him stay and eat is likely a battle not worth fighting though. I'd let him down in 5 mins if he regularly has no interest in eating at your regular dinner time.
hope this helps