You're welcome, on all accounts!

You certainly put more thought into the scenario than I did. I did review one of the other threads with what was more or less the same question I asked, but I wasn't really finding the answers I was looking for, and I wanted to hear from someone who was in law enforcement, like yourself. I appreciate your taking the time to answer my questions, thanks again for your service to our country and to your community.

This has been asked a few times here, and there doesn't seem to be a consensus, except among the video gamers (it seems). "Of course you pick up the AK and shoot your way out with a "superior" weapon". That's how it works in the real world of video games. If it doesn't work out, you get to try again and again, until it does. The reality of actual combat is a little different.
Consider the weapons involved. Brian and Mike were armed with Glock 9mm pistols. These are the weapons they trained with, and they seem to be relatively proficient with them. They are much less powerful than the weapons they were facing, but that does not mean, necessarily, less effective. Firepower was quite even. Although the bad guys had more powerful weapons, and more ammo, they were likely poorly trained. The bad guys weren't wearing body armor that I could tell, so even a pistol shot could be just as lethal as shot from a carbine. The opposition weapon was primarily an AK47 variant. These are heavy (compared to a Glock), and require BOTH hands to operate with minimal effectiveness. That specific AK was being used by a gangster-type, who are not typically known for proper weapon maintenance, or having more than a basic familiarity of their weapons. When Brian encounters the gunman in the hallway, each of them probably had about the same amount of time to react. At the time Brian first shoots him, the gunman had enough time to fire, and was positioned to do so. I'm not 100% certain, but I think I hear a metallic, mechanical sound that could be that of someone squeezing the clunky trigger of an AK (I reviewed that part a few dozen times). Likely meaning, the weapon malfunctioned, or the gunman was improperly trained. That may or may not have been intentional (scripted). The two scariest sounds involving firearms is, a bang that shoulda' went click, and a click that shoulda' went bang. Either of them can mean death. If that's what happened, that is why Brian (at least) is still alive at that time, and it may have some bearing on their choice.
Officers in a shootout are certainly "allowed" to use whatever means necessary to survive in this situation, including the use of weapons from assailants. It isn't something they were taught during training, but it isn't prohibited. Brian was likely familiar with the AK platform due to his military experience, but Mike was probably not. The situation was desperate, and they had one goal at the time: escape, and not getting dead doing it.
In my opinion, taking the AK would not have improved their immediate situation. Their focus was narrow, and picking up something that would hinder movement didn't fit the need. Mike may not have had an interest in learning a new weapon platform at the time, and Brian's left (support) hand was injured, as well. Add to that, the weapon may have been damaged or could malfunction at any time, as they often do. It is a very poor choice for a close-quarter weapon. It is more powerful, but under the circumstances, no more accurate than a handgun in a trained hand. High-power full auto carbines are a handful, under the best of conditions. They can be accurately fired when shouldered, but that weapon did not appear to have a folding stock, leaving a very heavy handgun that requires both hands to fire and may not be reliable, which was imperative at that time. Time was certainly a factor, and they didn't know what was waiting for them around the next bend. Changing defense strategies in that way could have seen them both killed, within seconds.
I don't know why that choice was made, but it seems to me to be the correct one at the time, considering the totality of circumstances. Through the course of that event, look at the score. Seems to me the pistols were significantly more effective than the opposition weapons. I just don't see how adding an uncertainty would have improved their situation. Ultimately, the barking dogs did more to harm them than anything prior.
If that were me, would I have picked it up? I dunno. Probably not. But I wasn't there.
Life is tough, but it's tougher when you're stupid.