There are some foreign doctors here in Japan, and you can do specialty training with your country's medical license, and without a Japanese Medical License. Supposedly they are going to make it even easier for foreigners to study here:
"The clinical training program covers non-Japanese who are licensed as doctors in their own countries but are not qualified to practice in Japan. They are permitted to train and provide medcial services under the guidance of doctors at designated institutions for up to two years."
Here is the whole article:
Also, they have talked about getting rid of the requirement for foreign doctors to take the Japanese license test:
You already said you know that you must be fluent in Japanese to study, so I assume you are taking care of that. Looks like you can do training without the Japanese license, but unless they change the rules, if you want to stay in Japan and practice, you need to pass the license test too. Good luck!