Can you create a unique invitation link for each survey participant that brings them to the same survey? If so, can you then assign dummy tokens to each unique invitation link?
I have had what I think is a similar issue, in that we had an open survey which then needed a unique number, and I use the term unique loosely here, that could then be used as a token in a subsequent survey so we could link the responses together.
This was done as the screener survey was "open" and the follow up surveys were all closed (also used as a way of letting clients send out invitations with their own customer ID's).
To do this you have to have access to the database and create a PHP file that generates a random number, populates the token table in the database with that random number along with a dummy name and email and then redirects to the survey with that dummy token in the survey URL. This allowed me to export the complete data and use it in a standard way for the follow up surveys with an ID that linked all the data.
PHP code to do this is below .. the Config.php should contain all your usual connection strings etc. for database connection