It appears that this was a glitch that could only be fixed by restoring my game back to the point where I entered the Universal Brotherhood. I'm not sure what caused it, but I repeated all the same steps and then it worked. So if anyone else runs into this, I recommend going back to a previous save. Or just shooting the turrets.
I also had a similar issue with letting my team in the back door. I saved right before doing it, and the first time I did it, it worked. Then I ended up reloading after a bit to that save and trying again, and that time (and every time after when I'd reload) the dialog would get stuck open and basically the game would get stuck there. That one got fixed by going one save previous to that. So it sounds like maybe saved games can get into weird states on this level or something.