In my opinion, your initial description ("a Japanese person who cannot speak English well") is probably the most straight-forward and concise manner in which you could have expressed that idea. Your other examples do not work for the simple reason that "Japanese" cannot be used as a singular noun. Other than that though, the last two options sound fine to my ears.
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KhoffPaco - I would particularly avoid one of your suggestions, "a poor English speaking Japanese." To me, it sounds like an impoverished Englishman who is speaking Japanese.Indeed! Thank you for the advice, Khoff.