Gloomyfates on Twitter (sorry, Elon) has been tumbling down the rabbit hole and has discovered that Ice Climber was distributed as a Controller Pak with a unique sticker in a giveaway by Nintendo Dream magazine.
"I can't find anything about this beyond the original magazine itself – no listings, photos, screenshots," adds Gloomyfates. "[The search term] アイスクライマー コントローラ パック only brings up a tweet from a user who seems just as confused, who found the pak pic. I can't find anything about it on tcrf. Nookipedia doesn't mention it on their page for Doubutsu no Mori under controller pak, the page for the NES games, or the page for Ice Climber itself."
Gloomyfates adds that they have "searched for hours, but the limited distribution of 30 winners just makes it impossible to find. My working theory is it works the same way as the unused ROM loading in the GameCube versions, just loading from the controller pak instead of memory card."
Animal Crossing modder @CuylerAC backs this up:
This is correct, Dōbutsu no Mori does not have Ice Climbers in its data. Dōbutsu no Mori does support loading external Famicom roms via the Controller Pak. So this is possibly the only documented use of that system. They did not use it at all during the GameCube era as far as I'm aware.
If you have one of these 30 paks in the back of your drawer, you could be sitting on a significant piece of Nintendo history.