If you are comfortable with your doctor. Please follow his advise. As this is best judged after a physical exam its only appropriate to advise on the situation looking at films, physical exam and diplopia charting.

Yes surgery is essential. This cannot heal on it's own as there is a displacement of the fracture piece and it has to be aligned in it's normal anatomy. One of your eye muscles is also affected by the displaced fractured segment and that's the reason for your diplopia and the muscle needs to be released surgically. Surgery cost will be decided by the Oculopastic surgeon based on the correlation with the clinical findings. There's no point in delaying the surgery as the prognosis worsens if delayed.