Tough one. A week between care days can be long enough as it is, so I think missing a week might make it harder. Can you decide after she has had her first session alone there? What does the carer think? She would be used to the settling in period and may know more about whether that one day off will cause issues or not.

My only concern would be if you took her for that hour, that she would pick on on the stress about the situation and connect that with going to FDC all the time, rather than it being related to the one off of it being her birthday. From you posts it has been a whole stressful process and I would be really concerned about adding more to it.