It sounds like your son is nearly in the exact same situation as mine... just finished half-day K and will be 7 when he starts 1st grade, completely ready for 2nd grade from the parent's perspective, similar size, prefers older children (particularly older girls), have the same resistance from the school admin to skip... Mine started off liking Kindergarten quite a bit but then about halfway through was clearly not learning anything academically. He's not to the point he's bored out of his tree, but if he were going to 1st grade next year at that school, I have little doubt that would have happened. Not only would he know most of the curriculum, he'd be at the school twice as long as next year (half-day to full-day). I can't imagine he'd cope with that well.
There have been numerous studies that indicate that grade acceleration does not have a detrimental effect on the social well being of the child. If anything, it seems like being too cautious about grade acceleration when the social aspect is the concern has a much greater potential to cause harm. A child should be adequately challenged in the classroom, period. If they are placed into a classroom where they aren't being challenged enough, they run the risk of becoming an underachiever, or possibly even someone who "checks out" from school. So from that perspective, I'd push for the skip to 2nd grade and not wait for the situation to worsen.
Don't count on the school to be proactive, by the way. They will likely be reactive and wait for the situation to get bad before response. And that response will probably not be, "Your kid is acting up... he must need a bigger challenge and grade acceleration." It will probably be, "Your kid is acting up... he must not be mature enough to accelerate a grade."
