I ended up booking on AA. I was using UR points through the Chase portal, and price/number of points needed was the same.
Now I'm not sure about the seat assignment though. It appears that you have to pay to choose seats in advance on BA, even when you book a Main Cabin (not Basic) Economy fare. That wasn't really clear from the Chase Portal, and there was no way to add on a pre-paid seat assignment anyway. I can pull up my reservation on AA's website using the confirmation # from Chase, but when I click to choose seats it redirects me to BA's site & the BA site can't find my reservation. I'll try it again tomorrow.
DW isn't going to be happy if we can't all sit together. It does say on BA's site that if you are traveling with children age 2-11 (which we are), they will allocate you seats a few days prior to departure. It also says that they try to keep your whole group together but in the event this is not possible, each child will be seated with at least one adult from their party.
Does anyone have experience with this type of thing with transatlantic flights? We're spoiled by Jet Blue where you get to choose your seat in advance for free. I don't really think I have a choice here though, unless I cancel the booking through Chase, book direct & pay for seat assignments, and have to cash out way more points to cover it. The flights were $2,300 but I only had to use 153,000 UR points through the CSR.