Congratulations on your pregnancy.
I would certainly talk further to both airlines before you travel.
BA state the following on their website:-
"For your and your baby’s safety you cannot fly after:
•the end of the 36th week if you are pregnant with one baby
•the end of the 32nd week if you are pregnant with more than one baby
After 28 weeks you must carry a confirmation from your doctor or midwife, such as a letter or certificate, in addition to your pregnancy record. This should be written within seven days prior to travel and confirm your approximate due date, that you're fit to travel and that there are no complications with your pregnancy".
AA state this:-
"Traveling during pregnancy
If your due date is within 4 weeks of your flight, you must provide a doctor’s certificate stating that you’ve been recently examined and you’re fit to fly.
Domestic travel
For domestic flights under 5 hours, you won’t be permitted to travel within 7 days (before and after) your delivery date. If you need to travel within this time frame:
â—¾Approval from your physician and a special assistance coordinator is required.
â—¾Your physician will be required to fill out a passenger medical form before your flight. (A special assistance coordinator will send the form directly to your physician.) "
I would seriously consider whether you actually want to travel at this late stage of pregnancy at all due to possible complications or going into labour early. You are also going to multiple destinations. Personally I would postpone the holiday and travel at a later date with your child.