We've had a few histrionic people described here, they really, really try to be the center of attention, often in the legal arenas and spilling over into the social worlds, often endlessly.
My ex has stated many lies and the evaluator rejected some of the lies but refused to reject other lies. Since she is histrionic, he should have considered her other comments as lies.
Maybe he's only discounting her claims that you were able to disprove with your documentation?
Is this custody evaluator qualified and well respected for prior evaluations?
My child has mild autism, and the evaluator felt stability in the mom's home is best right now. However, he will be re-diagnosed soon and think the mild autism is mostly a 4 year old acting out.
I had the impression that my preschooler had some developmental slowness, even slight autism around ages 2 to 3 since he wouldn't look directly at me and his talking was a bit young for his age. I attributed it to his dysfunctional home life since we didn't separate until he was almost 4 years old. From kindergarten to halfway through 3rd grade he couldn't sit still and couldn't concentrate in school or with homework. Teachers said he was bright but... . He didn't have a psych eval until after he was about 9 years old and by then he had greatly improved.
So it's possible that if your son's mild autism is caused or worsened by his exposure to a dysfunctional home environment, then the more time he can spend in daycare, school and with you, then the better he will be. Even if you don't get what you seek now, be consistent asking for as much parenting time as possible.