So, I can't update the game anymore... it was already in my library when the most recent update was released, but whenever I try to update I get hit with a "title hosted on an incompatible third party website"
error message.
That is explained in the FAQ page on the Wiki:
Itch App and incompatible third-party
Seems Itch App does not allow downloading from third party sites. Error message is “This title is hosted on an incompatible third-party website”. Game can be downloaded manually from the Download links on game Itch page.
Game can also be downloaded from Patreon and Subscribestar.