The walk-through susprise (No, the pedestal doesn't go with the sink and your guess is as good as mine as to why it was there!)

The sink today:


I have not even really cleaned it, as I'm afraid to. I do have some of that Barkeeper's Friend stuff as was recommended to me here, and I suppose for starters I should use that to clean it up a bit (?).
Next, I am wondering about using something like this to refinish it. Good idea or bad? Anybody used that kit before?
Confusing matters, I read that rather than refinish, you are better off having a cast iron sink or tub re-enameled & refired. There is actually a place near St. Louis that does that, but they aren't answering my emails. And subsequently (while waiting to hear back from them), I've read that because they no longer put lead in the porcelain coating, it's not that durable any more anyway, so it's not really worth the $$$ to have done.