There is nothing wrong with putting your birds outside. I don't know anyone here who doesn't.
Anyway I don't know where you're from but you can try finding the owner of that cat & talking to them about this problem. The cat owner should be more responsible. If he isn't, go to your local council & complain about it. As I said I don't know where you're from but here cat owners have a responsibility to their cats, especially not having them outdoors at night & if the owners ignore complaints we have a right to go to the council & they will do something about it themselves. We are not allowed to take matters into our own hands though.