Is the rust from the cover or has it eaten through the enamel of a steel bath?

No it’s not from the cover. It’s eaten through the enamel.

I’m not sure what you mean by new part. Also, reattach what fitting? Do you mean the plastic one that lifts the plug?

Also, I’m hoping to avoid taking the bath out because it will be a lot of cost.

It’s hard to tell from a photo but 1. Is this the drain at bottom of bath or an overflow whole in the side. Either way there should be a pipe connecting to it underneath. From this photo it doesn’t look like it’s attached.

It’s the overflow on the side of the bath.