I had the opposite problem as my son was skipped over 1st grade. He missed the chance to ever be a Tiger, but it was the right thing for him. The opportunity to be with his new classmates in the Wolf Den helped him adjust to the 'politics' of his new class.
I would recommend keeping him with his new class. Earning his Wolf Badge is only a fraction of the opportunities he can explore. He can earn a number of additional badges (75th anniversary, Leave No Trace, World Conservation...). He can also learn about leadership in helping his new classmates with their Wolf Achievements. I would even consider giving him a formal leadership position (such as Denner) to help keep him interested in scouting. He still gets to do things with his old 'mates at Pack meetings.
Good luck. There is no easy answer. The parents must be involved in the decision.