No, it's govt eugenics - they introduced 2 kids only rule. There are a couple exceptions - from the info you've given they don't apply here. The link Shell Helen Cave has provided in this thread gives full details.
I don’t think you get income support for more than 2 kids. If it’s still the drop off when the youngest is 5 though that should transfer to the new baby when your middle child is 5. You will get child benefit though. Double check with them though as I think if a child claims dla that age exemption don’t count or might be higher. Been a while since I claimed income support though so I may be wrong.
You can only claim benefits for your first two children the only additional money you get for more children is child benefit unless the third child had a disability and is entitled to dla then that is separate otherwise it’s only child benefit you can claim once the baby is born.