Thank you, thank you, thank you to all of you for the help!
For the life of me, I never saw that little tick box at the top of the list of our flights. I was able to reserve all of the first class seats I had hoped for.
On one of our flights (Zurich - London), BA doesn't offer first class, so we're booked into Club Europe. The seating system wants to charge me $32USD per passenger to assign seats at this time. So, I didn't book those seats now.