I would recommend to travel back to your country from Osaka in that case, so you will save time not going back to Tokyo, especially because you are only 5-6 days in Japan. A second reason is: if you stay overnight in Kyoto, you can wake up early and go visit Fushimi Inari Taisha or Arashiyama bamboo forest. Both places are magical in the morning when you are there alone and they don’t have opening hours. I went around 7 am, I was practically alone with my girlfriend, we heard the birds, we took great pictures and it was much more pleasant than visiting Kiyomizu Dera and Kinkaku-ji where we were slapped with selfie sticks,… Although Kinkaku-ji is also a must-visit.
It’s difficult to recommend the ideal district where you can stay. Shibuya might be quite crowded and expensive. I would prefer Asakusa. Maybe you can find a hotel with a view on the Sumida River. I think it’s pleasant having a room with somehow a view, which is rare in Tokyo. Sometimes I feel like a mouse in a cage with my 2 metre view towards my neighbour’s balcony and kitchen. Staying close to a subway station is also practical of course.