Any question answered...

To be re-finished correctly the following would need to happen:

Bathroom wall taken down, bath moved down to ground floor.
Bath packed and despatched to re-enamellers.
Bath shot blasted to remove existing enamel.
Bath re-fired/glazed at 800 degrees with fresh enamel.
Bath repacked, returned, re-installed.
Wall put back up.

That's £2,000 just for the enamelling part, leaving out the transport.

I reckon the drainage company will fight tooth and nail to avoid that, and it would take forever.

So I'd prefer to refinish the surface if I can, quickly and cheaply.

Seriously. Make them do all of this. And make them repair any other damage they make along the way. They have insurance.