Most manufacturers of non-metal tubs will recommend only liquid or bar soap and a cloth to clean the tubs. This is because harsh chemicals like Kaboom and abrasives will destroy the smooth surface and actually make cleaning progressively harder.
I would say that your best bet would be a non-abrasive mechanics liquid hand cleaner like Gojo. Don't get the stuff with pumice in it.
Just smear it around the entire surface with your hand and rub it around. Let it sit for a few minutes then work it with a rag like your trying to buff wax off of a car. It going to take a lot of work and sweat. Rinse with water and repeat as necessary.
I have found this to clean old stained toilets, enameled sinks and tubs and a variety of other surfaces without damaging them.
You can test a spot of the tub with a little bit first to get an idea of how the rest will turn out.