We eat about 80% organic on a budget of $700/month for five people - two adults, two preschool/kindergarteners, and one baby. I make lots of things from scratch like bread and babyfood and we eat meat 4-5x per week but the meat is not often organic because I haven't found much organic meat around here. I do quite a few coupons as well and I always have printable coupons for staples like milk, cereal, eggs, cheese, yogurt, and butter.
I shop at a military commissary where the food is at cost. If I were shopping in "real world" grocery stores I could see my budget needing to be at least $1000 for the same food. For example, the milk I buy is organic valley half gallon cartons. The regular price is $3.20 and I always have coupons for $1 off or $2 off two so a gallon of organic milk costs me $4.40. I know in some places that is what a gallon of conventional milk costs.
I also shop sales at coops if I happen to be traveling and can stop at one. A few weeks ago I found a sale on organic butter for $3.99/lb. and I had $1 off coupons so it was only $3/lb.. I bought ten pounds and it will last us a few months. I have found that by shopping sales I am able to get really good prices, often less than conventional, but it means stocking up then and there and having to store large amounts of food.
I also use amazon.com for things like crackers and pasta. They have good prices and free shipping when you buy a case, which is 12 boxes of crackers or 6 packages of pasta.
I know you said you don't care about grains but most of the wheat in the US is genetically modified and is heavily sprayed with roundup, the genetic modification makes it so the spray doesn't hurt the wheat when it's coated with it.
I have done an organic produce box club but it was $52/week for the smallest box which was not really enough produce to get us through the week. If you have the time and inclination you could look for an organic farm coop in your area that will allow you to exchange time spent planting, weeding, picking, and filling orders for produce. My sister did that and with one morning a week (with her little kids along) she was able to get the cost of the box down to $12/week. She did get some oddball stuff sometimes but I think it made her try things she otherwise would not have.
If I were trying to eat organically on $50 per month per person I would:
1. focus on the "dirty dozen" and similar lists that show the produce that has the most pesticides. After reading the list I no longer pay a premium for organic onions since conventional onion have very, very little pesticide on them and the outside is not eaten anyway but I am sure to never eat conventional peaches since they top the list. You may find that your favorite produce is low in pesticides.
2. start cooking from scratch. If you look at the ingredients in processed foods they are all very similar. It is the same few ingredients reconfigured a thousand different ways. For example, you can buy a package of organic flour tortillas for $5 or you can make your own (it's way easier than you would think) for less than $1. Same goes for things like bread, crackers, cookies, cakes, etc. I buy 50# bags of organic flour, with a computer you can look up tons of recipes to make with it and since it's whole wheat which has protein it can take the place of more expensive meat.
3. buy in bulk. This does not have to mean huge packages if you have a coop with bulk bins near you. You can get great prices on exactly the amount you need. Try googling "coop advantage" to see if one is near where you live, they always have bulk bins. If they have a sale on something you use lots of you can just ask for a whole bag. They can do it for things like rice, corn meal, various flours, raisins, etc.
4. try not to stress too much. When I first went organic I felt like it was all-or-nothing. I bought organic spices to replace my conventional ones, special sea salt, all kinds of organic grains I didn't know how to cook, and so on. I should have done it gradually and focused on things that eat a lot of first rather than nitpicking over things like organic pepper that we only eat very tiny amounts of. I would look at what you eat the most of and change those first then change other things when you are able to. I have found that we are much happier now that we have allowed some conventional food back in. We were so concerned with everything being organic that we were very limited in what we allowed ourselves to eat and it made us unhappy.
I think it will be super tough to do it on $150 a month but it is possible. Good luck!