Short answer: "Rather than Joan do it" sounds much more natural than "Rather than Joan does it". In fact, I don't see myself ever using "does" there at all! I need to think for a while to give you a longer answer to the last part of your question.
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Rather than Joan do it, I'd prefer to do it myselfIt sounds strange to me. For instance, what pronoun would we use instead of "Joan"?
I can imagine saying "rather than have Joan do it, I'd prefer to do it myself"; but the "bare infinitive" version sounds quite strange to me.