Kids who have a higher-carb diet *may* be more likely to tantrum because they are not getting their blood sugar/insulin regulated with a protein companion for the carbs. I am not saying, by the way, that your daughter's tantrums are exclusively based in a protein-lack, but it can't be a helpful contribution to the problem of tantrums.
When I worked at a daycare (two different ones) the food routine seemed to be 'you get what's on your plate, eat what you like'. We did not make special "I don't like that" concessions: either you ate what came on the cart, or wow, that's up to you not to eat. Granted, your daughter's daycare may heavily rely on 'lowest common denominator' foods like dry cereal and goldfish or graham crackers, so be aware that she may be developing this sweet tooth partly at school.
At home, our way of feeding my son is pretty simple: you get what I put on your plate. I have a highly-restricted diet and honestly, some of what I like to eat, he just hates. I will often boil up four or five eggs at a time for a very friendly protein for him at dinnertime. It's my compromise between "I don't like fish" and no proteins whatsoever. But at each meal he gets some veggies, fruit and a protein on his plate, maybe a piece of whole-grain bread with butter or a small bowl of rice or pasta salad, but that portion only makes up, at most, 1/3 of what's on the plate. Because he mostly likes what I serve him (unless we are being picky that day, which I just ignore), he knows that he's welcome to have seconds on his favorites once he eats the other stuff.
A., there have been a scant few times we've had tears over food, but when they happen, he goes to his room. "Come out when you are ready to eat what's on your plate." or "This is what's for dinner." No negotiating. No apologizing. Just 'go get yourself together and come back when you are ready to eat.' I will not force him to eat, nor will I put out new foods. Sometimes, *before* I put food on the plate, I might ask him "which veggie *don't* you want on your plate?" but A. only if I've had previously-cooperative behavior that afternoon AND only if we've had ease in eating his food. We've done pretty well with this.
I wouldn't delve into Gluten-Free unless you are working with an naturopathic doctor who suggests it. If you want to do 'drastic', first go into your cupboard and put the carb snacks out of reach, away. When we eliminate the option entirely for the child, this actually helps. Or you can divvy up a couple bags of crackers into ziplocks for the next day the night before. Next, put out meals which are nutritionally balanced and then let her learn to eat them. No rewards. Any tantrums go straight to her room until she calms down. Kids like variety and dips, so you may want to consider a ploughman's plate for your picky eater:
Here's a sample:
Whole grain bread with butter
Some apple slices (1/4 of an apple, no more)
Carrot, celery sticks, red pepper strips
Small cup of peanut butter, small dollop of ranch dressing (for dips-- I find this actually keeps the meal more interesting)
Cheese slices, Hard-boiled egg (you may consider removing the yolk, some kids really don't like it)
Small portion of nuts or olives
I usually prep this while I'm making dinner for my husband and I. (I must cook my veg, Kiddo likes his raw, so it's only an extra second to cut up a few strips I don't cook). Hard-boiled eggs can be done in advance and everything else is ridiculously easy to put together. I like ploughman's plates because they are predictable for the kid and allow me flexibility. For example, instead of that piece of whole grain bread, do three small crackers with peanut butter on them as well as the fruit, veg and protein. Sometimes I'll substitute the egg with smoked salmon since Kiddo does like that. We don't do the ploughman's plate every night by any means, but when I am making something for dinner I know Kiddo won't like (too spicy, cooked veggies, etc.), this is a good way to *think* about putting food on the plate.
And if she's still waking up early and is being cranky, let her have a banana about a half-hour before bed. Not cereal or crackers, but a banana. (If she wants a nut butter with it, great.) I think, though, given your situation, that you could also empower yourself by asking the preschool teachers to keep a food-journal for all the snacks and meals at preschool that she eats for one week. Let them know that you want to know what she is actually *eating* and not just what's being served. This is to find out what she's readily eating when there are clear and consistent boundaries and limits. This will help you feel more confident putting some of those "I don't like..." veggies and fruits on her plate at home, knowing that she does, in fact, eat them at school.
And I second the suggestion of the Love and Logic books. I know many families who have successfully used their ideas/techniques for discipline and child-rearing. I will say this, too-- if tantrums are Alyssa's way of getting what she wants, because you respond to them with acquiescence, then you won't change that behavior until you change your own. I like giving a "yes and when" answer... "Oh, you may have more crackers tomorrow-- for now, you may eat what's on your plate. You've eaten all of your crackers for today." When she gets used to having limited amounts of her junk foods, things will get better. Good luck.