first i sympathise with you and have to say it is bad form when BA refuse to honour original seat requests during equipment swaps, and your carefully layed plans go to ruin. absolutely should not happen.

I have a stopgap solution here.

Get 1 adult split off the booking.

Then come check-in (and please make your OH stay away and not be known to the check in agent) time, your 4yo and yourself (+ infant) should be seated together (as its BA policy to seat children with parents).
It does not solve the problem where your other half will be seated away - but at least it gives you that little more space, which in my experience makes some difference when trying to tend to an infant in a bassinet seat.

Upside: if you OH gets a good seat without anyone beside them, maybe your other neighbour will be amenable for a swap!

Permutation: get your other half upgraded (pts etc) - this should increase the chance of your neighbour wanting to do a swap

it would be horrible (for you and your neighbours) in my view to get the solo bassinet in the middle seat. it would make like a bit better if you could get 1 aisle seat either side. having said that, it is entirely possible for 1 adult to travel with 1 infant (and another child too).