All the people that are saying "I took my DC out of school for 1 day, we had a lovely time, just do it" seem to be deliberatley missing the actual point of this.
The OP said herself that as her dd does all the after school activities, her DPs shifts are what they are and she has a high needs baby (what is that by the way?) then she is unable to spend this time with her daughter at any other time and that her dd has not had this one to one time for 15 months.
Therefore, I can only imagine that this situation is unlikely to change in the foreseeable future. So in 6 months, or 8 months or even in 6 weeks the OP will be in the exact same situation Wanting to spend one on one time with her dd because she "misses her"
So, as taking her out of school is the only time in the world ever that she can have this entire day of local activities its not too much of a stretch to imagine that it will happen again. And again.
Or that the ops dd will think "oh mum hasn't spent any time with me, I'll ask for another mummy day"
Those of you saying you did it for one day and it was fine....did you also have children doing after school clubs 5 days a week? And a partner working shifts? And a baby that prevented you from spending any time with your child for 15 months?
For me, the point is, if the OP decides to do this one day and not make any changes to her life, then she will always be in that situation.
And a day off every 6 months because you cant give up a club, cant find a babysitter or because, despite having a family holiday, your dd is apparently desperate to do local activities is quite honestly bizarre.