I do not think that the OP should ask for compensation, but I do think he should write to BA and let them know how their family-friendly policies impact his perception of their premium product. The airlines won't know exactly how many people are upset by this unless someone tells them.

I think there are a lot of people saying, "It's not my fault" or "No one can control the situation" when in fact what they mean is "I consider the solutions to these problems to be inconvenient/unacceptable/drastic."

There are two parental examples in this thread that I think are highly instructive. First the gentleman who said that, since 18 months to 3 years is the worst time to take kids on airplanes, they would not be taking their child on any long-haul flights during that time period. He is to be commended for a sensible attitude that surely makes everyone (especially the child) happier. It's not like the toddler is going to benefit from flying across an ocean and having experiences it won't remember. I do have a problem with the opposite attitude: I think it's sad that someone would bring a child into the world with such a selfish attitude. Having children requires either a lifestyle change or sufficient money to hire full-time-staff to care for the child.

Second, the parent who said the only time their child was fussy on a flight was the day after she had her MMR vaccination. She was probably running a low grade fever, and suffering general discomfort. There are drugs which can help the children have a sleepy, painless experience on a flight. Most of the time fussy children are literally in pain (ear pressure changes, low-grade fever or infection) which can be ameliorated with Tylenol (paracetamol). Benadryl (Diphenhydramine) can also be a great help for encouraging a child to sleep while relieving underlying symptoms such as congestion and motion sickness. This is based on discussions I have had with two different doctors who are parents - they recommend pre-emptive drugging of children on airplanes. But of course you should consult your own physician.

Some other gems that I can't resist commenting on:
The difference is that humans are acutely attuned to the cries of an infant. The purpose of crying is to provoke a reaction from adults, and biology makes it near impossible to ignore the cries of an infant. I would not ask for compensation if snoring kept me awake, instead I would shake the person who was snoring and ask them to change positions. I would probably be arrested if I shook a crying child.
How is it out of their control? BA sells the tickets.When I travel with my niece, I bring books and entertainment, and spend a lot of time picking up after her. I understand it's difficult to travel with children, but then again, everything is more difficult with children. Which is why I chose not to have children. <shrug> And you are completely right - after having dropped my niece off with her parents, I have the most sublime moment on my return flight, when I ask the FA for a gin and tonic.