"Won't hunt" + "Good stopping power" means defensive use, right?
In a semi-auto platform, excluding 5.56, your choices include The commonly available .300BLK, 7.62x51, 7.62x39. All three come in a wide variety of platforms. All three have good stopping power.
.300BLK has the most versatility; subsonic low recoil loads and supersonic high performance (relatively) loads. Not cheap but not terrible either. I bought a bunch for under .50/round
7.62x51 is the highest performing but heaven help you if you shoot that indoors. A bit more expensive than the .300 BLK but a wide variety of excellent performance ammo available.
7.62x39 is very cheap to buy, does as well as the .300 BLK supersonic.
What kind of platform are you looking for? Rifle? Pistol? Suppressed?
Your choices in New York are somewhat limited for semi-auto. I don't even know if suppressed is an option.