If you think something is wrong, then you have the right to insist on seeing the doctor. You're paying. You don't have tell them everything when setting up the appointment, or you can tell them whatever you need to, to get in there. If the rash is not improving, and even if it does not seem to be bothering him, if it concerns you, then you have every right to get it checked out.
Heaven forbid DYFYS get wind that your child had a rash for 3 days and you didn't get it checked out, for whatever reason. Not that that should be a driving force behind parenting decisions, it just goes to show how backwards the systems are.
Ultimately, you are the parent and you are the client, so if one doctor won't see your child, then you can take him to someone who will and let the doctors office know that. It may be nothing at all, but you have the right to have your son looked at.