Is it permissible to sit and read the Qur’an in private, with each individual reading a part of it, intending to complete the entire Qur’an in the same sitting, or is this action an innovation (bid‘ah)?

The mentioned act is impermissible. A person is rewarded for what he recites himself or listens to for the sake of benefiting. The reward for recitation belongs to its reciter. These individuals are not considered to have completed the Quran. They should not do that; instead, either one of them recites while the others listen, or each of them recites independently without being tied to the recitation of another.