BA will not generally move people out of seats unless they must do so, e.g., one parent with the children. Equipment swaps are part of life and occur on all carriers (except the LCC's which don't have equipment to swap and simply cancel). Looking at your post, I am not sure what you expect BA to do in this situation. It has moved two other families requiring a bassinet into bassinet seats. In the pecking order for seats, they should come before your non-bassinet needs. You are free to vote with your feet, but you are not being fobbed off, you are simply asking for what can only be done by inconveniencing others and in commercial aviation, it is about the best for the most. Either you sit together and hold the infant (pretty much standard on most carriers) or you split up and everybody gets a decent if not perfect seat.

Your options seem to be to move one parent + infant to bassinet. Other parent with the two older children to the back. Many people travel single with 1-2 children and it can be tiresome, but hardly a serious burden. But, you know your children and we do not. The two parents can also swap during the flight if that makes things easier.

Alternatively and depending on your destination, e.g. Washington or Baltimore or perhaps elsewhere, you could look at availability on services to IAD or PHL and ask BA to make that change for you. BA may or may not want to charge a fee here because it really has no obligation beyond what it is doing. Changing to IAD or PHL may have logistical challenges which exceed your BWI flight, so need to consider those in the mix before even asking.

As others note, if you stick with BWI, simply see what can be done as the flight approaches, at the check-in counter and at the gate. People cancel, misconnect, and simply no show.