In case anyone is also having this problem in the future (as I just ran into this myself) -- in response to Durka531's suggestion:
The place "where the script kicks in" is not the gate/door/mirror itself, but one of the little swirly portals on the floor. For me, the script kicked in when I Fireballed Avernus/Sophia to either of the two swirly floor portals that are closest to the gate/door/mirror. Idk if the script will kick in if you Fireball them to either of the two swirly floor portals furthest from the gate/door/mirror.
When you blast Avernus/Sophia to the correct place, three demons will spawn in front of the gate/door/mirror. If using Avernus, a dialogue with him will suddenly begin -- the same dialogue in which he asks once again if you're ready and whatnot. But this time, once this dialogue ends, the demons are already spawned and you can actually commence the fight. If using Sophia, no such duplicate dialogue begins -- you just start fighting. Yay!
Thank you, Durka531, for this solution! Much easier/faster than running all the way outside, travelling to another location, travelling back, running all the way back here and hoping it works. :)
I'm closing this thread to stop it from being necroed.
