Camelback makes cleaning kits with brushes included. Any liquid diswashing detergent is pretty basic( as in not acidic) and should be like the bleach with out leaving the odor. I hang mine up inverted and keep a paper towel half in/out of the fill hole. It acts as a wick to draw the little bit of water left in the bottom out and keeps it open to air dry.
Or try a decent tequila.
edit- go with plan B
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