Is it permissible for a husband to refrain from intimacy with his first wife if she agrees to it for the sake of childcare, as a condition for marrying another?

It is not permissible for a woman to request the divorce of her co-wife. What your last wife stipulated – that you divorce the first wife or keep her in a state of suspension – is impermissible and you are not obligated to fulfill it. If the first wife is given the choice between divorce and remaining married, and she chooses to remain, willingly waiving her right to conjugal relations, then there is no sin upon you for not having relations with her. However, if she changes her mind and demands her right, then you are obligated to give her her right.