Accidentally discovered a solution of sorts. May not apply to all other people suffering from the same issue but…
I was looking at the task manager and noticed a “World of Warcraft” background process was present, despite the game itself not being actually open and running. I killed that task and the stuck update download was able to proceed as normally.
I didn’t notice the process because I had the processes in alphabetic order so the “World of Warcraft” background process was way at the bottom of the list of processes.
I’m back in-game now. Salty af over the stupidity but I’m glad it’s over with.