I've got the opportunity for him to shoot a doe over a feeder between 10-30 yards. Good setup for a 10 year old with whatever we go with being set up on a field pod.

Option 1- 300 blk 110 Barnes TACTX Supersonic 16" barrel

Option 2- 223 62 grain nosler partition. 10" barrel suppressed

Option 3- 300 blk 208 AMax subsonic 16" barrel suppressed

Option 4 - 9mm 147 grain hydro shock 5.5" barrel subsonic suppressed

Max shot distance is 40 yards.

Flame away

Why can't he decide which he likes best?

No.

Option 1.

Option 1

Provides the greatest margin for error.
Remaining options require increasing levels of skill, and honestly, I'm still not willing to trust any of those subsonic loads to anything but a perfectly placed head or neck shots.

Option 1 provides your son with the highest probability of success, even with less than perfect shot placement.

My vote would be option #1. I have been using a 16in 300 Blackout with supers to hunt deer with for the last 3 years and have never looked back. Just remember it is for him, so I would properly educate him on the pros and cons of each and see what he thinks as well. Just my 2c. 

Option 1.

30 cal supersonic at 40 yards would be a good choice.

I'd go with option # 1 too.
Never been impressed with any subsonic load on der/hog size game.  The wound channel looks like an ice pick wound in most cases.

#1. As already stated if those are the only options or has to be AR-15 type.

I don't know of many states that would consider the 9mm a legal caliber  for large game hunting.  

With that said I would look at the rule book and see if there is a caliber restriction of any kind.  For example here in NV we have to have a centerfire caliber of 22 or greater that will generate a minimum of 1000fpe at 100yds.  Other states I have lived and hunted in have a minimum of .243/6mm caliber restriction.  

I would make sure I didn't set my son up for failure on his first hunt.  I would make sure whatever rifle he was going to use that he was proficient with it shooting from multiple shooting positions and ranges.  
I would also want to use enough gun that if he did make a less than ideal shot it would have enough bullet diameter and energy to penetrate through the animal and leave a good enough blood trail that I could track.  If that means that you use reduced recoil ammo for a larger caliber to practice with and then full power ammo when the time comes then so be it.