There is no way to directly control the secondary Ice Climber, although some of the alternate costumes will feature Nana as the lead Ice Climber, including the white costume that you mentioned. Popo will always appear to be the lead in the character portrait, but Nana is actually the lead for every other costume (shown second, fourth, and sixth below):
As the Ice Climbers character, the player controls only the lead Ice Climber, while the secondary Ice Climber mimics the lead. It is possible to grab a different item with each Ice Climber, and possible to de-sync the two Ice Climbers in order to chain attacks or grabs together in rapid succession.
The best way to get the secondary Ice Climber unstuck is to reunite the two of them so that they can move together again.