Update: I don't think this is being caused by itch. I think it's something to do with Brave (or my Brave settings) and the fact the game is loaded in an iframe, as I've repeated the process on Edge and haven't had the issue on it for a good while (and it was pretty frequent). I've updated the OG post in case someone stumbles across this post via Google or something lol.
The problem appears to be this “ephemeral site storage” feature in Brave.
https://brave.com/privacy-updates/7-ephemeral-storage/ (2021)
You can change the ephemeral storage setting by going to brave://flags and toggling the Enable Ephemeral Storage option.
https://dev-pages.bravesoftware.com/storage/ephemeral-storage.html