Our kids have been paleo since birth (which I'm sure makes it easier). They still love to go on carb binges (fruit/juice/veggies), but if the choice is meat or nothing they have no problems eating a good steak for dinner.
You wouldn't let your kids touch an open fire, or play with electricity, so don't let them eat poison. It's up to you as a parent to ensure your kids safety and guide them in the right direction. Kids respond well to boundaries if you are serious and stick with them. Just don't give in to them (and I know it's hard when they whine or use their cute eyes on you).
A few bad nights is worth the lack of arguments from then on.
The "Seven Deadly Sins"
• Grains (wheat/rice/oats etc) . . . . . • Dairy (milk/yogurt/butter/cheese etc) . . . . .• Nightshades (peppers/tomato/eggplant etc)
• Tubers (potato/arrowroot etc) . . . • Modernly palatable (cashews/olives etc) . . . • Refined foods (salt/sugars etc )
• Legumes (soy/beans/peas etc)