Hm... I've tried doing that, but that seems to be specifically for the installation of the games themselves. I think the SquirrelTemp/ and Roaming/itch/downloads/ folders are specifically internal folders that can't be moved. The downloads/ folder in particular is just filled up with zip files with numerical names (123456.zip, e.g.), which I'm assuming are the packages that are then unzipped into the folders defined in Manage Install Locations.
Rather than changing the install location of my Library games, I was wondering if these internal folders could be installed elsewhere.