Oh well, if it is being a bad mother to put your school age child in after school care, I must be on the verge of being a DREADFUL mother, since my four yr old has been in nursery from 8am-6pm on weekdays since he was two years old!
Do I think he has suffered? Nope. He settled brilliantly, and from the first week, he used to be upset that it was closed on weekends, and now he is older, he runs straight in to see his friends, and I regularly have to sit and wait for him to finish whatever activity he happens to be taking part in.
I teach in a secondary school, and I am lucky that the nursery is next door, but to do my job to the standard I expect of myself, I am in work from 8.10 until 5.45 every week day, I do work for 3 hours every night, including weekends, and I work at least 50% of my "holiday". As a consequence, the time I have with my son I make the most of and appreciate, much more so than when I was at home with him full time, and he benefits from engaging with children his own age.
It's not always the best thing for a child to be attached to their mother's apron strings until they are prised off to go to school at 5. I know when my son goes to school he will settle quickly, and I will be able to leave him in a good quality after school club, whilst I earn to make a good and comfortable life for my family. Not to mention keeping my own identity and sanity.
I don't have a problem with anyone who wants to be a sahm, but I do have uses with people who tell me I am a bad mother for working full time.
OP, you would not be unreasonable to ask at all. If you don't ask, you don't get! Good luck!