I don't trust that the information about my identity would not be recorded while selling me my "free speech token". So the chilling effect on free speech would be exactly the same.
What if the digital infra that issues the token is state or Federal software? That should significantly reduce privacy concerns?
That would largely depend on the implementation details I think. Both those of the ID check itself as well as the precise nature of the tokens.
Consider a somewhat extreme example. A preprinted paper ticket with nothing more than a serial number on it. The clerk only visually inspects the ID document then enters the serial number into a web portal and hands it to you. When you go to "redeem" it the service relays the number back to the government server rather than your local device doing so directly. That would be far more privacy preserving than the vast majority of present day clearnet activity.