I think the use of a private pediatrician will depend on your insurance plan. We have a worldwide plan, and wherever we live customer service provides us with a list of PCPs/GPs/specialists etc and we self refer, or visit the GP if we are not sure what the problem is, or if it is just for something simple. However, some plans are for specialist and hospital visits only so you will have to find out your particular coverage.
Otherwise it shouldn't be a big deal to register at a different surgery. We lived in a village in Cambridgeshire which had a surgery and dental practice but chose to register in the neighbouring town after recommendations from friends.
Edited to add: Please know you have my sympathies over difficulties with your GP. Without going into detail we had problems with ours when my daughter was small and had a life threatening illness. It took a move to a different surgery and much insisting on my part that she was indeed ill. Wishing you all the best.
