If you need to go under 10", .300 is the way to go for sure. It's designed for that. But make sure you understand you're getting something on the low end of the intermediate round spectrum.
If the build is just for fun and you don't already have a 5.56 SBR upper, just do it in 5.56 and make it 10.3" or longer. If you do already have a 5.56 and don't have a particular use, I'd suggest looking at what ammo you want to shoot out of it, comparing cents per round, and picking a barrel length to match the round.