Would you change anything if you were building an ~$800 AR?
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Personally, just about everything. I also don't know where you're seeing those prices on MOE bits or the Hogue grip, but those are far lower than what I've ever come across. And don't forget shipping and transfer fees.
BCM is good to go, but there is no good reason to buy a 16" gun with carbine gas unless you plan to shoot Tula .223.
If you're sold on BCM, I would do
this , then cut the FSB off and finish it with a UTG Pro 15" SS hand guard. But you're looking at $750 on the upper alone to go that route.
Right now, today, I think
this upper from PSA is pretty hard to beat. Throw in an $85 BCG and charging handle combo from DSA, you'll be at $546 after shipping on your upper.
Then I'd go with
this lower build kit on whatever lower you choose.
Just finish it off with a set of MBUS sights, and you'll be at about $820 shipped. Regardless, you definitely should think about a FF hand guard from the start, and you will also quickly realize that you should have sprung for the CTR stock over the MOE.