Hi,

One of my Patreon tiers is extra content. All my games are free. Is there any way to embed that extra content on Itch, and have it locked behind a Patreon login?

Or, am I stuck embedding it in a post on Patreon itself?

Our Patreon integration works is by letting people claim ownership of a project by being a patron of yours. 

I can't think of an easy way to accomplish what you want with the same project page (both free and restricted content), without using some price tags. It might be best to have a separate page that is "Restricted" mode where you have the files for your Patreon supporters.