I had my first VBAC in a private and my second just last week in my public hospital. Both were unbelievably supportive of the process. Your c/s seems like simple bad luck ( my second, first VBAC did this and pushing for nearly 2 hours sucked !!)
I'd honestly investigate the hospital policy at the public and private options you would use and go from there. I found my public hospital to be better in all honesty however this could of been due to already having a successful VBAC so some of the restrictions were not imposed on me.
Good luck!