I am in the process of building two ARs, one a .300 Blackout pistol with a 12.5" barrel for deer hunting in Illinois and a normal 16" rifle for my brother. I have the rifle lower (mostly) finished, but am waiting on money from my brother to build/buy the upper. I have already ordered most of the parts for the pistol. If the lower receiver remains in it's current state and I build the .300 blackout upper, could that be considered in constructive possession of an SBR?

Not if you have a pistol lower, or even a virgin lower without a stock attached.

You are good.

Centerfire rifle hunting in IL?

The ATF says it's a pistol, and .300 Blackout meets the ammo requirements for Illinois. Put a bob sled or a modified magazine that only allows you to load one round and you have a single shot pistol that is Illinois legal.

What does the STATE say it is?
Many states are pretty far behind.

According to numerous people on The AK Files, Reddit, and even a thread on here, so long as the AR pistol uses at least a .30 caliber bullet with either a straight wall cartridge or a bottleneck cartridge no longer than 1.4 inches and has a magazine with a single shot follower, a bobsled, or a single shot lower receiver, it is legal to hunt deer with in Illinois.

AR Pistol Hunting in Illinois

AK Files IDNR Letter

I would never rely on "numerous people on The AK Files, Reddit, and even a thread on here" for
legal advice that could have serious consequences.
Like a short barrel rifle charge at the STATE level.

They are NOT required to defer to BATFE and Federal laws, though some state actually do.
You want to see the relevant statutes, and if it is completely clear, relevant case law.
There are a few attorney here, and they do give good comments.

I am not one, but a PhD EE that put his wife through law school.
I was her 'study aid' and so I red and learned just about everything she did.
I have read MY states (Virginia) gun related laws and BATF rules and regulations.

From my email to the IDNR:

"Mr. B

Provided a legal handgun caliber is used and the firearm is converted to single shot only, an AR platform pistol is legal for firearm deer season.

-Sgt. Stuart Fraser, #505
1 Natural Resources Way
Springfield, IL 62702"

That is information coming directly from the state that I can verify and corroborates the letter posted elsewhere, so I'm confident that what I am building is legal.