Report Builder: User Prompt for Database Connection

We have report servers that get copied monthly for a specific snapshot of a database at a point in time. Each one has a separate user name and password. 

I have multiple reports needing to point to the report servers, but I need the user to be able to specify which version (so is it the October 2023 copy or the January 2023 copy, etc.).  So when they pull up the report in the service, I need it to prompt them for a user name and password that they can input in order to point to a specific copy of the database. Does anyone know if this is possible? I tried parameterizing the login but the gateway didn't work with it in there and I know that has to match exactly.