Porcelain enamel (true enamel is glass bonded to metal in a kiln) sinks have an iron core.
If it's chipped to the dark glass, there are epoxy patch kits that can be color matched by mixing the supplied pigment or that have pre-mixed pigment in the epoxy. Apply after thoroughly cleaning the damaged area with alcohol to remove any oils.
Don't expect it to be visually perfect.
If it's chipped to the iron and you have rust, forget it as a DIY and send it out to a porcelain enamel restoration expert.
Body filler is porous to water, etc. Automotive enamel is a crosslinked polymer that doesn't have the hardness necessary to withstand the abuse unless it's properly baked.
Also, there are newer tub repair services out there that work on porcelain enameled thin sheet metal tubs and showers. They use an acid etch to roughen the enamel surface and an epoxy based paint to apply a new surface to the tub. You might see if a dealership for this is operational in your town.