I agree there is a good chance they will cancel off one of the bookings as a duplicate.

Do OLCI and PYOBP for the first one. This way, if they do cancel one of the bookings, it will be later one. If you have a chance of making the earlier one, you can make a dash for it, armed with the PYOBP.

As soon as you know you're not going to make the earlier flight, do OLCI for the later flight (if it lets you), but do not yet PYOBP - it may be useful to do that later, so you can go back into OLCI and be sure that the booking is still good.

Then call BA and get them to offload you from the earlier flight and cancel this booking for any refund of taxes (if anything due). Ask them to check the later flight, that it hasn't been cancelled. If it has, they should be able to reinstate it.

Then go back to OLCI and PYOBP for the later flight.