Well, depending on how many potential courses that a first time user would have access to, you might be able to accomplish this using groups. Groups allow for up to 50 conditions, so if the course ware available to new users is less than 50 then you can create enrollment conditions for each course, so if a user is marked as complete for any of them they are automatically assigned to this group. Then set up a separate Welcome Badge with the Assigned User set to be this group. People will only be added to the group once, and the badge assignment will also only happen as they are added to that group.
I’ve created similar situations, but not this exact one so it’s obviously worth testing first.