Yeah, I tried the .243 20" barrel setup in one of my Armalite AR10's...purchased new as a .243. Great shooting cartridge, and I still have a bolt gun in that caliber for which I reload. I ended up shooting my 20" .308 Armalite AR10 way more even though it's not as "fast" and "flat" as the .243. I swapped the .243 barrel for a 16" pencil Armalite .308 to get a carbine AR10 and love it. The .243 barrel is sitting in the closet and will probably remain there. I love .243, but it's mainly in a good bolt rifle. I agree with you trying to talk him out of it. Not a bad AR10 cartridge, but there are better semiauto cartridges out there for the big AR's.
Edit: I'll add something else here. His wanting a 16" barrel .243 is not generally considered anything close to optimum for that cartridge. Even in the new Ruger Precision Rifle in that caliber, a 26" barrel is the way to go. I'm not saying that one needs a 26" barrel to make .243 work, but it's very obvious it's a longer barrel cartridge for good performance. Anything less than 20" in an AR10 .243 is probably a waste. If he wants 16", the .308 is a cartridge that does quite well in a shorter platform.