.223/5.56 pros and cons
Velocity wise, a 5.56mm M193 round will deliver around 2400 fps in a 7.5" barrel. Commercial .223 in a 55 gr load will be closer to 2300 fps. The 62 gr M855 will do about 2350-2400 in a 7.5" barrel.
You need around 2700 to get a .223 FMJ to tumble, and while you can achieve that velocity in a 10.5" barrel, any way you slice it the muzzle velocity will just not be sufficient to make that happen in a 7.5" barrel. Consequently, you'd want avoid shooting FMJs in a 7.5" barrel PDW. However, I've found that most .223 soft point bullets will still expand reliably at those velocities, so terminal performance is still very good - if you stay away from an FMJ.
On the other hand, in terms of noise and flash, as you've noted the muzzle blast and flash on an unsuppressed 7.5" .223 is obnoxious - and is definitely not something you want to fire in an enclosed space without hearing protection.
9mm Luger pros and cons
The 9mm Luger is available every where and if you build a quality 9mm AR pistol it will feed hollow points just fine. You've also got lots of options from sub sonic 147 gr hollow points to 124 gr and 115 gr hollow points, all of which will expand, and all of which will be effective out to about 150 yards.
Accuracy with my 10.5" LAR-9 has bee superb - for a 9mm pistol or even a 9mm carbine, with 5 shot 50 and 100 yard accuracy between 1.5 and 2.0 MOA with jacketed hollow points and about 4 MOA with inexpensive heavy plated bullets.
A 9mm AR pistol is also a lot less abusive on unprotected ears, in the event you ever have to fire it indoors without hearing protection. Overall, I think a 9mm AR pistol is a lot more pleasant to shoot than a .223/5.56mm pistol, indoors or out.
9mm Luger is also less expensive and allows for more practice for a given dollar amount.
Of course, if you decide on 9mm Luger, you then have to choose the magazine pattern.
Olympic arms used to make lowers for Sten pattern magazines back when they were common and very cheap. Then they switched to a modified Sten mag that was not cheap, and eventually Oly switched to a polymer mag that had the same profile at the mag well as the .223 mag. They obviously were not as easy to store or carry.
Colt uses the Uzi pattern magazine with an insert in the magazine well. Rock River Arms also uses the Uzi pattern magazine, but has a dedicated 9mm lower.
The advantage with the Uzi magazine are that Uzi mags are still cheap (but supplies are drying up) and you can modify them by adding a cut for the AR mag catch on the side, and a clearance cut on the back for the ejector. They are also very tough and very durable. They are however primarily limited to 25 and 32 round variants, although there are some Uzi pattern 10 round magazines available.
Glock pattern magazines are the third option, and 9mm AR-15 lowers that use Glock magazines are available from several companies - which might be nice if you also own a Glock pistol. Some folks also prefer how pistol magazines load as opposed to SMG magazines.