If your goal is 22-24% protein, and your chick starter is already at 21.8% protein, then you probably don't need the game bird starter. Just add a little fish (or really, any kind of meat -- chickens are omnivores and can eat anything -- just cook it for safety issues) and you'll boost that protein up easily.
As far as what type of fish, it probably doesn't matter, as long as it's not rotten. I'm in the Pacific NW, so there's lots of variety here. I eat a lot of fish, and I typically give some leftovers to the chickens as treats. My main breed is Red Dorkings, which are an ancient English Breed. They were around before preformulated chicken food, and were perfected when English farmers fed them scraps and free ranged them. English farmers in that era had a lot of fish, and journals talk about finely chopping up the fish skin, and boiling the fish heads and throwing them out to the chickens as their main protein source. I have pecking issues with my Dorkings when they're not given meat protein (it doesn't matter how high the vegetable protein is, they need meat), so that's what I do. You could probably buy fish heads, or maybe even get them free, from your local fishmonger, if you have someone local, or from a large supermarket. You could also probably use koi food. Just don't overdo the proportions. You only need to boost up the protein a little. You don't want them to grow too fast and get joint problems. If they're freeranging and eating a lot of grass, which probably only has 2-7% protein, depending on where you live and what type of grass you have, then you can give more fish than if they're not free ranging and only eating 21.8% chick crumbles. It's all about the proportions.
Regarding organic vs non-organic foods, I don't want to open up a whole can of worms here, as I know there are very strong feelings about this. Here is the way I look at it. I'm not saying this is right or wrong, so don't send me hate mail like someone did last time. If you disagree, that's fine. But in my opinion, organic is great (I have a 10 acre organic farm), but organic has nothing whatsoever to do with being a balanced diet. And for growing babies, balance is critical. I wish that there was an organic diet out there for my chickens, cats, dogs, etc that I had confidence in, one that I knew was tested and had the perfect nutrient blend. I'd pay double for it. But that doesn't exist. I'd love to support the companies that have all the right marketing labels on their products, but I can't support an organic diet that doesn't support my animals adequately. So for my chickens, I use mostly Purina products, mix in a little NatureSmart (an organic product from Cargill Animal Nutrition, but the birds don't really like the taste as much as the Purina, and I'm still on the fence about it's adequacy), and free range, plus daily garden and/or meat scraps. I use whole oats as grain treats for the birds that I don't want to gain weight, and a local 5-grain plus fish oil scratch grain mix called Winner's Scratch as a grain treat on the birds where weight gain is ok. As far as which Purina product I use, that depends on the age and breed of birds, and how fast I want them to grow. Heritage turkeys need 24-30% protein until 8 weeks, then 20-24% 8-20 weeks. My "high protein" dual purpose chicken breeds (Dorkings, Sussex, Buckeyes) I like 22-24% protein in starter formula, plus a tiny bit of meat, which probably boosts it up another 2-4%, plus free range, which probably brings it down about 4%. I'll customize that a bit if I'm raising them with poults, or keep it like that until the cockerals are ready to change at 12-14 weeks, and the pullets at 18 weeks. I have decreased the total protein down to 18-20% at 8 weeks before, and haven't liked the results as much (this is limited to Dorkings, and is likely different in other breeds).
One thing that is really important to remember is that chickens are VERY adaptable. You're only likely to get into serious trouble if you're feeding only a single food during a critical growth period, and it's not properly balanced. Variety allows for imperfection in feeding, as long as you're meeting their basic needs. Now, if you're feeding to maximize growth or profit or breeding quality, you need to be a bit more focused on the details. But if you just want a nice healthy farm flock, you can be a little flexible, as long as you don't go to far afield.