I had this exactly as you described, turned out to be my sacrum and not my coccyx.
This physio gave me a number of stretches and this one helped immensely:
https://youtu.be/UZHUCUdTXfE
It’s unlikely your gp will do anything for any kind of back pain other than pain meds so if you can afford it I can’t recommend enough seeing a physio/ sports injury clinic.
