I had a similar one. The jacquard fabric can show wear pretty easily, but yours has held up well and isn't too dirty, so I would avoid the full dunk if possible to avoid "unforeseen events." If the strap were immersed, there might be some stiffener inside that would wrinkle, for example. The edge coating might start to peel off.
I would first try spot cleaning the stains in the lining and applying conditioner to the leather. Then decide if dunking will help, despite the uncertainty and potential complications. Have fun!