The sentence is wrong. There are two possible ways of saying this. Note the plural 'women': Feminism is not a stick with which to beat other women. Feminism is not a stick to beat other women with. The speaker should not have put the extra 'with' at the end of the sentence. This is a clear case of someone mispeaking. It's actually very common in natural, unrehearsed conversations to make slips of the tongue of this type.

In general in English you can repeat the same word in a sentence if it has different meanings/uses in the sentence. You can do this even when you repeat the same word twice in a row. In your sentence the with means the same thing each time and it’s definitely redundant to use it twice. Simple good writing would say to eliminate the second one because the sentence is clear and understandable without it. I don’t have any definitive rule to tell you it’s a mistake, but it sounds like a mistake and that type of mistake is common in spoken English. We don’t always perfectly align the end of a spoken sentence with what came before due to limits of thought and memory.

For example, I say: I knew you live "in" New York City. Did you know I knew where you live "in"? I hope I could help you understand it.

Interesting. I think the wrong statement sounds as good or better than Su.Ki's right ones. Partly it has to do with the whole phrase "beating someone with a stick" which is very old-fashioned and beyond the realm of modern experience. So a sentence ending with "beating other women" is not going to involve sticks or physical beating 99% of the time. So doubling up kind of reinforces this unfamiliar use.

Chris indeed knows what he is talking about.