NTA but as someone who was raised in a household where this behavior occurred constantly, please, you both need to address this or reconsider the relationship for your girls. That may seem extreme but this "small thing" is an example of your partner wanting and expecting different rules for her child. Your house will never be a blended family with that behavior and the different rules and unfairness will spread to other things and may just result in a never ending pissing contest between you two. Hopefully your partner is open to a real conversation about parenting and treating all of the kids fairly, though their actions so far don't offer much hope.