Lucy,
You ask "what would you do?" Phrased that way, it was easy. My private insurance situation did not permit me to absorb extra costs (not old enough for Medicare, no VA, just ultra-high deductible coverage as an employee). I had to stay in network, period.
As to choosing a doctor, if economics allow, choose the one with what you deem to be adequate experience, and with which you are comfortable. I had to research within my network plan, and have had no issues.