I am having a similar issue. I have a project in "restricted" access and I've generated a download key. I have the itch.io app on one PC and I successfully used the key to install the app. I then installed the itch.io app on a different PC, logged in with the same user from the first PC, and tried to use the same download key after publishing an update and I get the invalid game error. The game updated as expected on the first PC.
We have a few testers I'm trying to document this process for and I'm not sure how to continue as it doesn't seem to work after the download key is used once. Even though I did NOT select the option to allow the key to be claimed. My intention is to use the key for a few testers and not have to send them updated keys all the time.
Thanks for any further info you can provide!
-Mike (Numinous Games)