Is a husband permitted to obey his wife in severing ties of kinship with his half-sister and her mother (his stepmother) due to a past incident not intended to cause harm, and are the stepmother and half-sister considered part of the kin who must be shown kindness?

If your wife saw something suspicious, or if your joking with your sister contained lust, then you must repent and not return to such actions.

If there was no suspicion or temptation, then visit your sister and her mother and do not sever ties of kinship, for you are not obligated to obey your wife in severing these ties. Severing ties of kinship incurs the curse and wrath of Allah, as confirmed by His Almighty saying: "So would you, perhaps, if you turned away, cause corruption in the land and sever your ties of kinship? Those are the ones whom Allah has cursed, so He made them deaf and blinded their vision." And the Prophet (peace be upon him) said: "No severer of kinship will enter Paradise."

Your sister is a relative whose ties must be maintained, and it is recommended to maintain ties with her mother. Your wife's threat of divorce is not a justification for severing ties of kinship. Rather, wisdom must be exercised, and you should try to convince her, or maintain ties of kinship without her knowledge, or reconcile with her in a way that does not lead to a break or separation.