Success!
I finally had time to get the gear back out and attempt the repair. The parts arrived last Thursday and unfortunately, I had a full weekend.
Most of this is a record in case someone else comes across this post and needs hints or details.
Desoldering was tricky but straightforward. Somewhere recently, I heard someone say that the secret to desoldering with a manual pump is to add solder, and that worked a treat. I melted the existing solder and then probably doubled the total solder. Going back with the iron and sucker, almost the entire joint got removed on the first suck. It's also much easier to simply clip the leads of the bad part. This reduces the heat sink, and makes it possible to pull each lead individually as they are released.
I finished with some braid until the holes appeared clean and round.
At this point, I reassembled the PSU and tested it without the SCR, as suggested. The open-circuit voltage was reported as 12.6V, as expected. I did not test with a load, but I wanted to see A) if something else was broken or B) if the replacement caused further issues.
Replacing the part was much harder. There is an insulating board between the PCB and the part itself, and the insulator holes don't align well with the holes in the PCB. In the end, inserting the part at an angle and then standing it up got the middle lead underneath the insulator and into the PCB hole.
I probably used too much heatsink compound, but I'd rather clean it up than add more later.
The heatsink held the part nicely in place for soldering, and the actual solder job was anticlimactic. It's not my best job, but it's nothing I'm embarrassed about. The photo is before I clipped the leads.
Reassembly and testing appear to report a fixed unit. It operates both in CV and CC mode.
Thanks everyone for your help.