Have you asked about the possibility of having the private results accepted instead of having to wait on additional in-house testing from the district? We had our youngest tested privately because he wasn't "far enough behind" at school to qualify for testing. Once the private results came back, I made an appointment with our school's diagnostician, shared the results with her and asked if we could use the results to being the process for getting him into special ed. She took the report to the district; they reviewed it and accepted the findings. It saved us several months and the risk of different results complicating things.
As far as the actual IEP went, it was a collaborative process between the diagnostician, the special ed chair, a special ed teacher, his current teacher, and myself. I found that the school actually wanted a very liberal IEP in elementary school so that they would have the freedom to try what they wanted to see what worked.
At this point they are asking me to try an alternative option, but getting them to accept a private option they'd have to pay for? I think that would require a lawsuit.
You may find that the school isn't willing to pay for an out-of-district option until it is proven an in-school IEP is a failure.
