Does it restock at varying intervals? Or is it like the token shop, where they announce it periodically?

its random, can take minutes or hours even

not every restock gets posted about, but if you look in General you will see a thread about restocks of this shop. Reading quickly through it will give you a feel for how often it restocks and when the last one was.

Me, I randomly stop by the shop multiple times while I am doing other stuff here. It doesn't make me a huge profit, but it does get me a whole set for Achivement before it retires. This desultory pattern also means I almost never run out of Invites when I need one.