You don’t say which sector you are working in. Family friendly shifts only work if they do not affect the bottom line, an organisation with everyone happy with their working hours but going bust is no good to anyone. If you can make a good case as to why it will be to your employer’s advantage, you may get a better response e.g. you are available to work at peak times or you find someone else within the organisation who is happy to work the hours you do not want to etc. otherwise, it is probably better to change jobs.