I agree with the rules and the behavior that is tolerated. I know you want them to eat, but they will not starve if they miss a meal. When they are done, they are done. Dinner is a time for talking and sharing, for laughing and talking about the day. It's not play time but it can be fun time in the sense of talking pleasantly and enjoying each other. Don't make it a battle ground. If they leave the table, they are done, and clear their plates. Don't argue, don't lecture - just be calm. I would let them come back one time to sit and try again while you are working on this - and give them the same plate you took to the counter or put in the fridge. Don't start a new meal for anyone! After that, they are done. Don't get into the dessert war - it just shouldn't come up that they won't get "rewarded" with treats if they sit through the "punishment" of a healthy meal! Tell them you are clearing their plates and dinner is over. If they complain about being hungry, just calmly say "It's a shame that you didn't eat during dinner. Maybe tomorrow you can try again."
He behaves at daycare because it is expected and he watches other kids do it. At home, he does what he's allowed to do. It can be challenging when kids are tired later in the day, but as long as you are having dinner at a reasonable hour, it shouldn't be too hard to adapt.
Try worrying less about whether they eat - don't be so concerned about one meal that you give them all kinds of stimulation (TV) to keep them there. They will make up the calories and nutrition the next time. If you can keep calm and not buy into the tension with your own sense of dread, it will be better in a maximum of 2 weeks, I promise! Maybe sooner! You are setting the stage for later years too, so the sooner you adjust your guidelines and expectations, the better it will be!
We have a 7 year old in our extended family who has a weight problem, cries if he doesn't get a plain bagel every night for dinner, who won't eat a fruit or a vegetable or a bit of protein - and the parents give in because they don't like the tantrum. He will have diabetes before he is 10 - he already has metabolic syndrome. So set good habits early!