With an IEE, the school pays for the eval & the final product is the eval report. It is on the parent to pay the evaluator to attend the meeting where they go over the report. I've not seen a school pay for this.
It's not FAPE if the parent has to pay for part of the cost of the IEE. Ask the district for names of people who do psycho-educational, OT, PT and ST evals. See how much their suggestions cost and if it fits the budget they gave you. This is one eval with 4 pieces. It's going to cost more than an eval with only one or two areas that need to be assessed.
IEPs are not one-size-fits-all. What they cost isn't going to be one-size either. A student with more complex needs will need a more complicated eval and complicated means it's going to cost more.