If it was me, I’d soak in warm, soapy water before using an old toothbrush to clean out the nooks and crannies.
Another option might be a jewellery cleaner bath - available from Amazon.
I would not soak the link band. I would use a soft bristle brush, a warm microfiber dampened cloth and a toothpick if needed for the hard-to-reach areas. Then thoroughly dry with lint free cloth.
This is similar to what Apple recommends to clean the link band:
Just not sure about the sb version.
I wonder how the adjustable links are made inside as a sonic cleaner would be nice, but not sure how the vibrations could affect the internals if the band and the sb coating.