While the parent case for 300BLK isn't 223/5.56, you can make 300BLK brass out of 223/5.56. I'd go with 300, because it's an order of magnitude more common than 7.62x40WT. I'm not saying that it's inherently better, just that there's a big advantage to being able to buy commercial ammo.
6.8 is another reasonable option, as is 6.5. I'd go with the latter due to superior long range performance (although if you're considering 300 etc I imagine you aren't looking at that). Also, cheap plinking ammo is available.
Cost, convenience and reloading components favor the 300blk. It has really taken off and has earned a spot on the shelf at Wal-Mart next to the venerable 30/30. The 300blk shines in its intended role as a suppressed sub sonic round. .... and is no slouch using lighter supersonic rounds, being a decent 200 yardish deer cartridge. The 6.8 is a better long range choice but I rarely see it even at gun shops and it is more money. That 7.62x40 is a rare bird. It doesn't appear to bring anything to the table that the blk doesn't so..........No.