To this day, I don't buy into that crap. There's no way the CPU on the 3ds couldn't handle it. I'm sorry, I just don't buy it. I'm not really a fan of Ice Climbers, but it still doesn't make much sense to me. I'm no programmer, but how does Rosalina & Luma work so smoothly, and then Ice Climbers become impossible to create? Meh. I don't buy it.
Considering the stricter limit of players at once on the 3DS version (compare Multi-Man only having up to 5 fighters at once vs. home console iterations of the series handling more than that), it's likely due to processing issues.
The 3DS could handle more players, but that would come with frame-rate drops, which always remain a steady 60 overall, which would mean that lag due to not enough RAM would be a huge stain the quality of the game.
There's a reason Assist Trophies and Poké Balls only run at 30 fps and you can only have one at a time. The number of items at once is also slightly more limited than the home console games.
Next, Luma reacts to inputs directly from the player and is component of Rosalina. It's a small model that runs dependently on the player's inputs. The Ice Climbers, however, are effectively two characters running at once, with Popo being the main player and Nana being a special AI program. The fact Nana is as demanding as a player and runs her own AI requires a lot more memory and resources than Luma does. Having several players as the Ice Climbers would likely cause frame-rate drops as a 4-player match with all players using the Ice Climbers effectively amounts to 8 players on the stage.
My theory is: could the Ice Climbers work on the 3DS version? Yes, they could. Would they work smoothly and not degrade the game's quality? Not really.