I think I would have designated shoes for inside. Designated slippers are worn inside possibility you could have slippers that have a good sole if you can find some that provide the support you need.

Find a good slipper that you can carry with you. There are restaurants / chain izakayas / hotels that supply slippers, but they tend to be on the small side. Bringing our own slipper may be admired for its ingenuity, but that would be only reaction. Just make sure your inside never touches the outside.