Hi. I rarely steam fish, so my bamboo steamers do not smell like fish. I think your proposed method sounds fine. I have used dish detergent and brush to clean my bamboo steamers before, rinse with running water thoroughly, and then steam them again to clean.
If you worry about detergent, then you can do duo cleaning with baking soda solution first and then diluted white vinegar solution second, and then finally steam the steamers with diluted white vinegars. The two steps cleaning of baking soda and vinegar should cover both the basic and acidic cleaning. The final steaming with vinegar can get the vinegar to deep places where regular cleaning cannot penetrate. The steamer will smell like vinegar. If this bothers you, then you can rinse it with water and steam it with water to remove the vinegar residue.