Maybe it's a bit late, but I found a way to get another shot at this cursed sidequest.
First, in oder to avoid the doors to glitch and become blocked, you need to do the Rats in the Lab sidequest before any other.
If you didn't do this and the doors are permanently closed, you can still manage to get the quest. Do the following:
- Fast travel to the Jungle (I guess any location will do; I did it on the River Village) and go to the Game Lobby (in the Start button options menu).
- Look for someone who's playing chapter 11 and join that session.
- Once inside, play through the story until you get to the lab and do the Rats in the Lab sidequest first of all.
- After collecting your reward, wait for the checkpoint to complete, and then you can quit the session and your main save file will have the quest updated.
The doors should now remain open. If you get kicked out prematurely from the session or if the other players leave the session, don't fret, for it doesn't matter. Just keep playing until you reach the lab and play the sidequest. If for some reason you have to leave the session, just select "No" when you are asked if you want to load your solo checkpoint.
This should do it.