Because bears and wolves could compete for resources, bears might avoid areas that were commonly occupied by wolves, and young bear cubs have been documented to fall victim to the occasion wolf pack here and there, albeit the Japanese wolves were the smallest of the common wolf breeds.
I am not trying to say that Japanese wolves would solve the problem entirely, but I do think it is important to not try to sway ecological systems too much. Our interactions with nature can sometimes have unpredictable downstream effects.
If Japan were to be rid of bears, then I imagine their next problem would be sika or wild boars.