When the school agreed to a mid-year skip for my son, it was contingent upon the school psychologist meeting with him and providing her thoughts on his maturity, ability to cope with change etc. I am quite surprised that the school has asked for a psychiatrist evaluation and that they are paying for it. That seems a bit excessive and Master of None might be right- it might be a way for them to say "No and here's why." I think you may want to meet with the psychiatrist in advance and find out what s/he is looking for, what the goal is and get a feel if the goal is rule in acceleration or rule it out.

Honestly for us, the school psychologist became our biggest ally. She fought hard for my DS long after the skip and continued to push the school principal.