The 300 is not going to kill deer any better than the 5.56 with a premium hunting load like the Barnes TSX .223. The 5.56 is also easier to hit with at longer ranges because it has more velocity and less drop. I'd get the best 5.56/.223 I could afford and forget about the 300 which is basically a special purpose round developed for suppressed AR style weapons replacing the original 300 whisper which wouldn't fit the AR platform.

This is an AWESOME question because I have EXACTLY this!! I got the ATI Omnimaxxx 5.56 from here and then built a 300 AAC BO on my own and it takes 15 seconds to change them. I started reloading and cutting down some 223 brass to make my own 300 BO ammo and it's great!
The only thing I would recommend is to get another charging handle and bolt carrier group eventually as that is the only pain (not a big one) but at least the same one can be used for both. Also be sure not to mix up ammo in your mags!!

Although the AR-15 platform is not my favorite (I lean more to the AR-10 due to a better selection of calibers for the type of shooting I do) Having multiple uppers allows you to spend a bit more on the lower and have the exact stock and trigger you want no matter what caliber your shooting Example for the AR-10 I am working on having four uppers one in 6.5 Creedmore, one in 308, one in 325 WSM and one in 450 Marlin, this allows me to use the same basic gun for a large variety of uses. Like wise You can have Uppers on your AR-15 for 5.56, 300 Blackout, 6.5 Grendel or 6.8 or 458 Socom or 50 Beowolf. where building a complete gun for each caliber would cost you much more to get the same quality in triggers and stocks, though I do use a heavier stock with a bit better recoil pad on the AR-10 when using the 450 Marlin upper.

When it comes to ar15s I like to make sure the barrels are quality. I always look for "chrome lined". Another thing I look for in 5.56 is that it has a 1:7 twist, this will help stabilize those heavier bullets for more precise long distance shooting. As for the two calibers mentioned, perfect for plinking varmints on the 5 56, and the 300 is just as much fun but adds more knock down for larger animals like deer. ONE other benefit to the 2 calibers is that they use the same primers if you reload, and when it comes to switching uppers the barrel is the only difference. So you can buy one upper cheaper without the bolt and then transfer your bolt carrier group back and fourth.