Do you think that he can prove that "Whatever knowledge is attainable, must be attained by scientific methods; and what science cannot discover, mankind cannot know"? If so, please talk us through it.
I'd say that 'science' broadly speaking, is the pursuit of knowledge. And 'scientific methods' are whatever seems to work effectively in this pursuit (maybe Russell is arguing in the opposite direction and defines science as an activity
that conforms to a particular methodology?)
If it were somehow possible to discover absolute rules of ethics then they could form the basis of a science of ethics. Russell's point is that we can't have a science of ethics (or aesthetics) because the subject matter "cannot be intellectually decided at all, and lie[s] outside the realm of truth and falsehood." Perhaps he is echoing his colleage G. E. Moore here and regards ethics as a rather mysterious thing that we can say little about (I'm pretty sure he isn't being a relativist).
I don't think it's a denigration of ethics or aesthetics to say that they are not concerned with knowledge. Scientism is not simply the belief that science is the only path to knowledge (which in any case needn't involve a prescriptive claim about what science is). Scientism also involves the elevation of knowledge-seeking above all other human activity; the belief that all other human activity is intellectually worthless or meaningless. This is not necessary or desirable in my opinion.
To talk about 'truth' in art is only metaphorical. Art shouldn't give answers (art that tries to is trite), it should be above that. But no one thinks that all judgements about art are impossible, that we can't value some things more than others and that we can't have intelligent discussions about such things. It's the same for ethics. A "ten commandments" type list would be dumb. And Peter Singer is dumb. But there are non-dumbed down conversations we can have about ethics and they aren't a waste of time even though their purpose is not to discover truths about the world.