I have no idea but think you'll probably end up with a better result of you consider changing it to black or navy. That way you won't have to worry about potentially clashing with interior surface.

Because it’s the exterior you don’t need to resurface as such. You can just paint. http://www.castironbath.co.uk/blog/painting-cast-iron-bath/ Here is a UK link where it is more common.

Whether you DIY or get a professional it's not going to be cheap. To get a professional finish on it (Smooth and glossy) it will need to be sprayed on with a professional paint gun, the same type as used to paint cars. It will also need a resin-type of coating. Personally, I would contact those companies that do kitchen and bathroom resurfacing. They have all the equipment and do it for a living. They will be able to give you a ballpark $$ to budget.

If that bath is actually acrylic rather than cast iron it may be more cost effective to just buy a white acrylic version and sell the existing one to someone else. These baths come in around $1400 in acrylic.

Turns out now that I’ve been able to inspect it in real life the agent misled us - or he just hadn’t actually checked himself - but it’s an acrylic one!