Is vowing to wear the head covering (tarḥah) for life considered a vow of obedience that must be fulfilled, or is it an permissible vow that allows for expiation, especially given the difficulty of adhering to it and the statement "A covenant upon me to You, O Lord" made at the time of the vow?

If the vow was expressed as: "A covenant I make with You, O Lord," and you repeated it, then it is a covenant with Allah, and it becomes an oath and a vow if what is committed to is an act of closeness [to Allah] and obedience; otherwise, it is only an oath. Since you did not uphold the covenant and removed your head covering, you are required to pay the expiation for breaking an oath (kaffarat yamin), whether what you covenanted upon was obligatory or permissible. This covenant is dissolved by your failure to uphold it, just as an oath is dissolved by breaking it. You are required to pay only one expiation, even if you repeated the word "covenant," because the reason [for expiation] is one.