I think that forced apologies, backing kids into situations where they feel like they have to lie to stay out of "trouble," etc., only backfire. I remember being treated like that-- I know you're lying, and you'll stay in your room until you're ready to tell me what really happened, and thinking how ridiculous the whole thing was because my mom KNEW what happened. I don't think it's the best way to teach them about the truth.
It can help a lot to just circumvent the issue of needing permission to get food. We changed the rule from "you need to ask first," to "only at certain times of day." My DD1 really wanted the control and privilege of being able to get her own snacks, but I didn't want her eating snacks right before meals- I work hard to prepare meals, and we have a limited food budget, and it's nice if everybody comes to the table with an appetite.
This just happened to use a few weeks ago, actually.
So what I did was to make a sign, with two side-- one side says YES, and the other side says NOT NOW. I hung it in the kitchen, up over the window where it's difficult to reach. Most of the time, it's hung on the YES side. If the sign is on YES, then kids are free to help themselves to the snacks that I make available-- and I make sure they know where the food is that they can have, and keep foods I don't want them to eat out of sight and out of reach. Then, an hour before lunch, and two hours before dinner, I switch the sign to NOT NOW. The rule is that if the sign is on NOT NOW, you cannot help yourself to food. If there's an exceptional circumstance, and you really feel like you NEED food, then you're to ask an adult.
That way, there's nothing to lie about. I'd say let me see what's in that bowl. Then I'd say, let's go see what the sign says. Oh, the sign says NOT NOW. Honey, you can't take a snack when the sign says NOT NOW. Here, let me put that away for you, and you can get it when the sign says YES again.
I wouldn't then follow that up with any further consequence, unless it became chronic. If it did, then I'd wait for a neutral time, and I'd sit her down and say, love, you know the rule about snacks, and a few times you haven't followed it. If you break that rule again, then I'm going to put the snacks away where you can't get them, and I'm going to be in charge of giving snacks for a few weeks. If you want to be able to help yourself, you need to follow the rules we've set. (Then I'd get real busy in my own head, figuring out where I would put the food to restrict access, so I'd be prepared in case she decided to rest my resolve.) Then, if it happened again, I'd do what I said before, and then I'd calmly go about the business of restricting access-- putting snack foods in an inaccessible location. Then I'd mark for her on the calendar exactly which day the snacks would "come back again," and remind her that when they do, she will have to follow the rule about the sign.
I think the best way to handle lying is to set up the situation as much as possible so that there's no reason to lie.
Re: the drama. Some kids are really just like that. In the above situation, for example, my DD2 could just be like, okay, mama. My son might cry a little. My DD1 would have a full-blown breakdown, complete with accusations of terrible unfairness, and give me to understand that I'd ruined her life forever. She gets over it. If I've made the rule clearcut and simple, and explained it to her thoroughly, then I know I've been as fair as I can be, and so I just offer a little sympathy for her disappointment, and then ignore any further drama. I don't attempt to discuss what happened until she's fully calm.