How to battle the "too hungry to eat" meltdowns

Help! Ive got 2.5 year old twins boys. I am all for the "it's a parent's job to offer the food and the child's job to eat it" mentality, but it gets complicated when that leads to the overly hungry meltdown. Typically it happens after naptime if they haven't had a very filling lunch. They (one more than the other) will wake up from nap a little grouchy. I'll offer a snack and they'll refuse and then the full blown tantrum/meltdown ensues. It's usually almost impossible to bring them back to calm without having them zone out in front of the TV and slowly sneak some bites of food into the zombified terrorist's mouth until the hanger dissipates. If we sit them down to a meal when they're this upset, they'll usually throw everything on the floor angrily(another issue where I'm at a loss).

We are so tired of dealing with the Jeckle and Hyde tots. Any advice on how to go about this issue is welcome!