Can I restore a bathroom sink with automotive filler and spray on enamel?
If the "time period" aspect is important enough, there are places where you can send your sink for a proper repair/refinish (if it's possible - hard to say for certain without knowing more about it) - they will advertise more (if at all) for the cast-iron tub trade. It will cost more than a new sink, most of the time.

A DIY filler/paint job has odds of performing over the long term so close to zero that I'm willing to say "it won't work" and be done with it.

You can probably find a new "reproduction" sink that will also look right for the time period, if you shop around enough. You may also be able to find an old one in good shape through an architectural salvage firm.