Is bending down to pick up something that fell during prayer considered a bowing (ruku'), and what is the legitimate way to pick it up in this situation?

Bending down to pick up something that has fallen is not considered a rukoo' (bowing) that fulfills the pillar of prayer. It is strongly disliked if the worshipper is standing. However, it does not invalidate the prayer if done only once; otherwise, it invalidates it because it is an excessive action.

As for when one is sitting, it is considered a minor action that does not invalidate the prayer.

It is preferable to pick up what has fallen after the prayer or while sitting, unless there is a fear of loss. If there is a fear of loss, it is permissible to move to retrieve it, and the prayer might be invalidated by such movement, with details similar to a case where one fears harm to oneself or one's property.