I have all the DLC and the game on my Steam account. It won't let me get my free GOG version.

Try launching the game on steam, close the game, and then hit the refresh button.

I've got the exact same issue. I own 11 games on Steam from the latest Gog connect batch. The 10 other games have been added to my Gog.com account just fine, the problem is just with Democracy 3. :-(

I have contacted support and suggest you do the same to let them now it is not just one user having this problem.

In my case, clicking refresh on the Gog connect page and waiting for a while solved it. I hope it does for you too.