Yes, they are required to see your BRP card in order to confirm you have paid the IHS surcharge for your new visa.
However, you should ever have been registered as a temporary patient in the first place and you do not have to keep re-registering... GP receptionists are known for not knowing the NHS rules for spousal visa holders and often tell people they don't qualify for treatment when actually they do.
I would try again with the GP and see what happens when you show your BRP card. You could also try contacting the local NHS trust (or whatever they are called now) to explain the situation.
If they still refuse to register you permanently and you can't get anywhere with them, I would recommend finding a new GP (if there is another one in your area) and registering with them instead.
Sent from my iPhone using Tapatalk