Online cancellation within 24 hours is usually easy enough, so no conversation to be had.

What I’m slightly unsure of is if BA systems might automatically cancel one or more of the itineraries before you have a chance to choose… particularly if there are any overlapping segments (that is, you’re booked twice or more on the same segment). I remember even quite awhile back now, possibly during the Iceland volcano thing many years back, I genuinely wasn’t sure what I’d need to do and had my agent reserve space on two different possibilities while we tried to sort things. Sure enough, one of them was automatically cancelled within less than 24 hours, and it wasn’t I who had picked which one.

Obviously awhile back, so I don’t remember just how it worked logistically, but I was surprised at how fast they surmised that I couldn’t possibly go on both trips, even though they had different destinations. My considerably more experienced travel agent was, however, not surprised as he’d seen it happen before.

The most solid method would involve ensuring no overlapping segments, and possibly leaving the BAEC number(s) out of the booking until you pick which one, and add it in later. But that turns into more work which may not strictly be necessary depending on the timing of your decision making.