Is it permissible for a husband to take back his divorced wife solely for the upbringing of their children, without his desire to cohabit with her, and would their marriage be valid in this case?

There is no issue in a woman who waived her marital rights returning to claim them; because she has the right to waive all or some of her rights whenever she wishes, and this waiver does not affect the validity of the marriage contract. However, if the waiver was stipulated in the contract, there is a difference of opinion among scholars regarding it. We advise you with the saying of the Prophet, peace and blessings be upon him: "A believing man should not hate a believing woman; if he dislikes one of her characteristics, he will be pleased with another."