I want to distribute a special, closed-beta build of my game to my Patreon supporters and others I manually give out keys to. Is this possible?

When connecting Patreon to itch.io, it seems I still need to set a price for the software, or choose to give it away for free.

Thanks in advance!

You can use "Restricted" mode, and keep the files free on your project page. Only people with download keys will be able to access the page. Then you can enable Patreon linking, which will let your patrons claim download keys.