Was it obligatory to spend the night with one's wife in the same bed after divorcing her for the first time with the intention of an irrevocable divorce (al-bain) and before her waiting period (iddah) expired?

We do not understand what you mean by "but with the intention of a bā'in [irrevocable divorce]." If you mean that you will not take her back during the iddah (waiting period), then she does not become bā'in by this intention. However, if you mean that she was divorced three times, then the majority of scholars hold that she becomes irrevocably divorced (baynunah kubrā), and it is not permissible for you to marry her until she marries another man.

Regarding your staying overnight with the divorced wife during her iddah, this is not obligatory for you. Scholars have differed on the permissibility of your staying overnight with her if the divorce was revocable (rajī):

Malikis, Shafi'is, and one narration from the Hanbalis: It is not permissible to cohabit with her or be in seclusion with her, because that leads to being in seclusion with a forbidden woman and harming her. Hanafis and the preponderant view among Hanbalis: It is permissible to enjoy her, be in seclusion with her, touch her, and look at her, with the intention of reconciliation or without it (with a minor dislike, karāhah tanzīhiyyah, according to the Hanafis), because she is like a wife during the iddah.