The lying part isn't ok, I would mention that to the pedi and see if they/ you can figure out why that's going on. I don't know how she does with describing emotions, but you could try asking why and explain that you will put snacks here and those snacks will be whatever you think is appropriate,etc.
As long as it's healthy food, I wouldn't worry too much about the amount. My youngest son basically eats from the time he wakes up in the morning until the time he goes to bed. He will also go around and steal his brothers' meals. He's almost 2, so there's obviously an age difference, but he eats larger portions per meal than his brothers' do all day combined. His weight is in the 20% and his height is in the 90%. I've decided he just needs the fuel.
Anyway, I basically have to make sure he has easy access to food at all times or he steals food or eats nasty stuff he finds on the floor. Can't wait until he's a teenager. I'm going to have to work a second job just to feed the kid. :P