Let me preface this by saying I don't have the experience that a lot of other people on this board have. I've only recently realized that my kids are advanced and am still in the "barely keeping my head above water while the school all but ignores the needs of my children" phase. But I'm with Grinity. I would b wary of anyone tells you he is not emotionally mature. Lots of kids tend to take cues from the people around them. My son went through a phase in Kindergarten when I actually thought he was regressing. Turns out he was acting like the kids at school. I put him in taekwondo, where most of the kids are older than him. Not only does he get along better with the older kids, but he acts more like they do now.
I'm not saying he should definitely be skipped - but I do think it is one option to consider. I would guess, though, that it would be more difficult to get the school to agree if the psychologist recommended against it.
When I went to my last conference, I went armed with actual scientific papers about why children should be challenged. (I realized a few days before that the college I attended still allowed me to log in and use their library resources as an alumni, so it was like Christmas for me). My son's thing is math, so I tried to stick to math-specific articles. But I'm sure there are just as many for any other subject, or just in favor of challenge in general. The administrators did not seem happy to see me with all of those articles, but they couldn't argue with them. I didn't get what I wanted, but I did at least make a little progress. Good luck!