I'm not sure I understand why "slugging it out" has anything to do with your co-pay.
A co-pay is preset by the insurance company. When you go to your PCP, you either have a co-pay or not. It depends on your coverage. If you have a co-pay for your primary doctor, that is paid at the time of your appointment.
If you go to a specialist (in network) then there is a different co-pay amount that is payable at the time of the appointment. IF you see a specialist that is not in network, then that's a different story. You will be billed for going out of network. If you have a procedure that needs to be authorized by your insurance company, you will owe a co-pay when the billing is done. If you are out of network and have a procedure, you will have a higher payment after the billing is done.
The problem with office and hospital billing in todays world is everything is contracted with an outside billing company. That's what I worked in for years.
When my husband died in October at the hospital, I'm yet to get the bill. I know how much it will be because I have kept track of what's been paid by my insurance (and everything has been) via the insurance's web site so I know I will owe the hospital XXX amount of dollars. I have not received a bill yet from the outsourced billing company.