Welcome to FT ^

Whilst I appreciate your situation, but BA are playing by their rules on this one (morally, it may not seem great), but they do not have to give you seating together as you are on separate bookings.

I would save yourself any frustration, and check-in at the airport as early as possible and your request will likely be handled to your satisfaction there. The ground agents will see sense (or at least they will see a group of 3: 2 parents and a very young child) and accommodate you. I would not rely on a call centre agent to help you.

I hope that you get the seating arrangements that you want

Also, I think that threatening writing to the DM or some such national paper is usually met with a poor response. I have had this when I worked in customer services, and when I heard this, I stuck to the rules of the organisation and refused to 'flex' the rules. It is not a great method of achieving something you want.