Since she's the one demanding what seems like an unnecessary change, at first glance, I'd side with you.
However, "refusing," which you're both doing, isn't showing great relationship skills. What will you do when a real problem arises?
Do you really want to "win" and have your partner "lose" if it matters a lot to her?
I'd tell her you think you're both being stubborn and that you would like to have a discussion and work out something you can both live with.
First, I'd try to discover why doesn't she want to walk the dog in the mornings now? Is she, perhaps, worried about being able to adjust to getting there in the morning and anxious now about the added pressure of walking the dog? Maybe too embarrassed about that to admit it? Or is there something else you're not aware of?
Next, consider what's the most you'd be satisfied giving up, if anything? Would you, for example, walk the dog on "her" mornings if she did something for you in exchange? If so, what?
Like that. Keep going until you both understand each other's perspectives and have reached a resolution. Remember you like each other.
Or, if she's often like this and depending on the rest of your relationship plus its duration, it may even become a step in you realizing she's not the one for you.
Good luck with it.