I'm pretty sure that BA don't allow name changes for tickets, so making up a name and date of birth won't work. Although my children are now older, I'm certain that I added them, after birth, to bookings made before they were born.

However, there's a chance that things have changed in the last 5 years - it's probably worth phoning to confirm that it is still possible to add a lap-infant after booking. In fact scrub that - Google is your friend:

Making a booking for an infant who isn't yet born

If you know you want to travel shortly after you’re due to give birth, you can make your own booking as usual. Please contact us after your baby is born and we’ll arrange for your baby’s ticket to be issued.

You must always have a ticket for your infant's journey.

Newborn babies must be at least 48 hours old and have been born without any complications before they can travel with us. If you plan to fly soon after your baby's birth, please check with your treating doctor that they agree to both you and your baby travelling.

If you have had a Caesarean section or if other surgery was needed during or after the birth, then you can travel after 10 days, with medical clearance from our Health Services. We would advise that your treating doctor approve your travel first.