I am a plumber now. I got the sink unplugged. Yea me!
This task was made unnecessarily difficult by the original plumber, who sealed all of the joints with plumber’s putty. A couple of master plumbers have explained that there is no need to use plumber’s putty on this fitting unless you just don’t know what you’re doing—or you want to make it hard on anyone who needs to clean the drain in the future. No putty went into the reconstructed joints.
And it even stayed dry once I put it the pipes back together.