I had this happen to me too (although I didn't drop the actual mission, I just left and reentered Donatu). Basically for some reason, if you respawn too close to the doorway to the next room you'll be thrown in there without being able to get back.
I wouldn't fret too much. The DOff is... very niche at best, pure fluff for most, and the mission itself is extremely underwhelming (I think it's literally all of ten minutes in length).
Add me to the list with this bug. They should give a warning that there is the ridiculous cooldown as I am sure other people are going to miss out on this DOFF character too by dropping this mission when they see that they can't complete it as that is what the Devs usually say to do when you have a bugged missions
It works okay to just leave (tp to ship) and re-enter. As mentioned above don't "drop" it.
Don't die in the room with the two turrets - it will bug out and nothing in the next area will be interactable. If you're a lower level toon and the gas is a bit much, make sure you die in the transporter room.
After getting gassed one or two times, it should let you shoot the terminal as opposed to solving the puzzle.
"I weary of the chase. Wait for me. I shall be merciful and quick."
Happened to me as well, I just left the system and entered it again (didn't drop the mission as didn't see why that would help). I then put on a EV suit and took my time on the puzzle. Still, would be nice if they adjusted the respawn spot slightly to prevent the issue :-)