Wow. I think we have the same son.
Mine is also a 2nd grader taking 4th grade math at AoPS academy (so in a classroom with other kids), and is SSA to 3rd grade math in his public school. My school district is so rigid that technically speaking, he shouldn't have been allowed the SSA, and he might not be allowed a 2nd grade skip.
For my part, he isn't leaning anything at all in his public school math, and the Beast Academy 4 is pretty easy for him. I'm not planning on pushing him ahead in the Beast Academy math, though, since there doesn't seem to be a point in being so far ahead of the public school math.
Instead, these are some of the things I've done to supplement his math education:
-Download or buy books of math contest questions, and have him work through these. Elementary school math olympiad (MOEMS) has some nice books. You can generally find math kangaroo tests online (at least the canadian ones). Many AMC8 questions would be within reach, too.
-Logic puzzles: My son enjoyed the logic section of BA 4B, and there are many apps out there with the same types of puzzles.
-Music lessons
-Beast Academy online contest.
Re: the reluctance for another SSA- Mine is kind of the same way. He feels weird about being viewed as different at school. The logistics of the SSA have been stressful to him, since the math class times are not aligned. Also, for each year we SSA, that's another year he will have to travel to the middle school for math and take classes with middle schoolers, which makes him very nervous. I wish schools had better answers for kids like ours, but they really aren't equipped to deal with outliers.
