I think there have been a lot of good ideas - I do think that the protein thing is important. Can you find sources of cheaper protein?
We eat on a budget, organic - here is what helps us. Price the things you use the most of, and shop for bargains on those. Esp look at things that are expensive - you can either find them cheaper, or eat less of them. For eating less meat, try soups, stews, stirfrys- things where the meat is cut into small pieces - it stretches it much further and gives you a good gluten-free option, compared to the idea of a plate with meat, starch, and veggies, or any option that involves a big hunk of meat as the main part of the meal. Also makes cooking easier and clean-up, too, if you eat leftovers - most any soup or stew gets a lot better the second+ day as the flavors meld - I'm really looking forward to leftover Belgian chicken stew with prunes tonight....and make full use of your meat - when I roast a chicken, we eat roast chick for 2 nights (family of 4), then either sandwiches or a BIG pot of soup with the last pieces of meat. Clean the carcass well and then use the bones to make stock with carrot parings, ends of celery, and onion peels/ends. Makes way better, healthier stock than anything you can buy. Even soup once a week will save lot of $$.
What other options are there in your area for cheap organic? - Trader Joes (meat, veg, actually they have organic in most dept's)? Grocery Outlet (cheese, pantry goods like cereal and crackers, inc gluten free), bulk food from a grocery or coop, or mail order coops like Azure Standard or Frontier (stick to sales stuff, esp)?
For beans - have you tried beano? Do you cook them yourself, soak over night and dump the soak water for new before cooking (prolly doesn't happen for canned beans/processed foods)? A piece of dried kombu (seaweed) is also supposed to help. A bottle of beano is probably about the price of a nice piece of organic meat. Maybe certain varieties are more tolerable than others? Worth a try, and beans are so healthy. Or probiotics?
Don't just look at the "dirty dozen" - look at the "cleanest conventionals": this list is best to worst:
http://www.foodnews.org/fulllist.php So onions and avocados are your best bet, as would be eating organic in season, growing some if you can (if you don't have a space, can you trade some labor for a friend's garden space?), pricing fruit, veg etc by the pound (I shoot for as close to $1/pound as I can for organic, if it's more than $2, I think twice and consider it a luxury).
What grains can you eat? Rice, quinoa (also a decent protein), oatmeal? And what do you eat for breakfast? Oatmeal with nuts or peanut butter, a chopped apple, and dried fruit is healthy if you can eat it, but if you can't, can your family eat it? Is your kitchen completely gluten-free, or can you cook some grains (even if not gluten-containing ones) just for them?
What are you eating for other meals? If they are all meat heavy, then yes, it is spendy....
I know it's a real pain to cook more meals, but if you can't tolerate grains, bean, etc, couldn't your family? Could you cook a big batch of something you like and freeze for you, and then eat a different meal, even if just for the main course, and eat the veg etc that they do?
Our costco has organic food - frozen veg, butter, milk, some meat - if the one near you doesn't, is there another one within driving distance that does that you can do a stock-up run at? (we have two near us, one has almost nothing organic, the other has quite a bit). They'll look it up for you if you call and ask - I got my organic turkey that way for T-day.
And here is the very biggest money saver I can give you, if you don't do it already - PLAN A MENU. We write up a menu every week, write up the list based on that, adding stuff for lunches, breakfast and snacks, and stick to it. Otherwise, you spend a lot of money, and walk out with two bags of food that won't make the dinners you need.
We spend $200/week on gas, groceries, and meals out. So we probably spend about $150 a week on groceries, including the money from our stock-up budget(for bulk orders, etc - those things messed up our budget too much to fit in the weekly money, as they varied so much - stock-up is $125/month, total). We do eat about 90-95% organic at home, and my kids are about the same age as yours, and we have lactose intolerance issues in the house, which is also limiting and expensive. (my 4 is a picky, light eater, too).
I think you could eat mostly organic, with some adaptation, esp on the meat front. I don't work, and spend a lot of time bargain hunting to keep the budget in line. I will say that gluten-free makes it harder, esp if you buy GF breads, etc - they are super $$$.
Please post or PM if you have questions - I do think that you can still eat a chunk organic, with priority given to some items. Sorry it ended up being a book....