If you really want to, go ahead, but I doubt the ASA will be much help:

- www.bmibaby.com does tell you that the flights are operated by BD and that you will be transferred to the BD website.

- BD does offer cheap fares. The fact that there might not be availability on the precise dates you want does not mean that the fares do not exist. If the fares do exist and there is a reasonable amount of availability (which does not mean plenty of availability), the ASA will reject the complaint.