Regardless of whether you check in or not, the adult passenger not making the flight will be treated as a no-show. It is unlikely that you will be able to claim a refund for any portion of the ticket in a no-show situation, but it's always worth a letter to BA.

Your strategy of checking in to protect the 4th seat will certainly "block" the 4th seat from others trying to select it at online check in, but once you get to the airport and the no-show is confirmed (likely at the bag drop, although your post is not clear on whether you are checking bags) the seat will be freed up for others and will certainly be used if the flights is full. Unfortunately, you cannot request the seat be used by the infant (or for that matter yourself) as a no-show forfeits any right to the seat (even by the others travelling on the same booking).