I dole out fruit (fresh & dried) and things like flapjacks, carrot sticks and houmous, occasionally a yogurt, the odd biscuit, crackers etc as snacks, usually something mid morning and something at 330ish (school).
Many 3 year olds need snacks in between meals - in fact I would say most of them, so cutting them out is not a great idea, usually.
Crisps etc are a treat, not a daily snack; ice lollies are a dessert (unless it's summer!)
I would restrict the treats, too.
To be honest, I would let him carry on with his 'one mouthful is enough' habit, but point out that he may feel hungry later. Learning about their own hunger and how to regulate it is the best way to get children to eat well. If he has one mouthful and says he doesn't want any more, say 'OK, I htink you will be hungry later though' and leave it at that. And when he does get hungry you can say 'do you think it was because you didn't have enough lunch?' (hand out a banana and a drink, but not lots of snacks)
Then you can move onto (when he's had his one mouthful/however much it is) 'do you remember yesterday when you were really hungry after breakfast/lunch/tea etc?' and gently get him to understand he is in control of how much he eats (of what is put in front of him) and you are in control of what is put in front of him.
I've gone a bit far past your 'what do you do about snacks' question, but I think it's all related.
I wouldn't let a 3 year old select his own snacks from the cupboards tbh - I would say 'you can have this piece of fruit or that cracker'. And I would regulate when the snacks happen - nothing within about 45 minutes of the next meal, more or less.