Advocate here — the most useful thing for your situation: what you're describing (a kid who doesn't fit any one box) is exactly what a comprehensive neuropsychological evaluation is built for, and that's different from an autism-specific diagnostic. A real neuropsych eval is organized around differential diagnosis — it tests cognition/IQ, academic skills, attention and executive function, language, and emotional/behavioral functioning, precisely to sort out what's driving the symptoms (ASD vs. ADHD vs. anxiety vs. a learning disability vs. some combination). So your instinct to push for "comprehensive, not just ASD" is the right one.
On Mt. Washington: they do more than ADOS-style autism testing, but before you commit, ask the scheduler directly — "Does this evaluation include cognitive, academic, attention/executive, and emotional testing with differential diagnosis and written recommendations, or is it autism-focused?" Get the scope in writing. That one question saves families from ending up with a narrow report.
Two things the private route won't tell you that matter a lot:
1) You can request a full educational evaluation from your public school at the same time, in writing, for free. Under Child Find the district must assess in ALL areas of suspected disability — cognitive, academic, speech/language, OT, and social-emotional/behavioral. It's a separate track from a medical neuropsych and you can run both; the school eval is what actually establishes IEP eligibility and services.
2) If the school does its own eval and you disagree with it, you have the right to request an Independent Educational Evaluation (IEE) at public expense.
The insurance-vs-private-pay point above is real — private-pay evaluators usually spend more time and write more tailored, usable recommendations. Whatever you pick, ask to see the recommendations section of a sample report. That section is where the value is; a diagnosis with generic recs doesn't help you at the IEP table.
You're asking exactly the right questions. Good luck.