So I've been working on that tool a little bit on the side every day since, and I'm happy to announce that the first image was just synced from a shared album on one immich instance to another album on my own instance ^^
It's far from finished, but nice to see that this works already, and it was indeed not hard at all.
Btw, my goal for this is to have a minimal proof-of-concept tool that works well for syncing a single album – not to implement a whole lot of functionality or friendly UI.
Using this for importing shared links would also be cool tho. Maybe I could turn it into a library or something.