How can I make an enamel-on-steel bathtub feel/sound more substantial?
Bed it in plaster (which is very likely what the tub manufacturer recommends - most do.) Foam can work but is not preferred (easier when you have an already installed tub, though.)

Bedding on plaster is literally get ready to go, test fit, dump a pile of plaster on the floor under the middle of where the tub goes, set the tub. Tub ends up solid, you don't have anything that's going to rot in there, and the plaster is easily broken out when the tub eventually needs to be replaced.

As for your cockamaimie scheme, the diving weights alone will cost more than the bag of plaster; and it's an ugly mess and waste of effort .vs. doing it right with plaster.