When a project is marked as free, users cannot “claim” it as theirs, but they can still download it. That’s why when the price changed to paid, any person that downloaded it in the past, does not own it now.
There is a tool on Itch.io that is specifically for kickstarter, to solve your problem. It is located under “Edit Game” -> “Distribute” -> “Kickstarter import”. Have you tried that?