Also, you should be aware that the coating on your tub is not paint. It is porcelain enamel. This is a powder coating that is electrostatically sprayed onto the steel tub and then baked at high temperatures so that the powder melts and the molten particles fuse together to make a glass-like coating that is very hard and durable. It is much more durable than field applied coatings like tub renewal products.
So, you CAN have your tub refinished, but don't expect the durability of the field applied coating to be anywhere near that of the original porcelain enamel coating. The field applied coating will LOOK good, at first at least, but it just won't stand up as well as porcelain enamel.
Really, for the long term, it might be best to patch the chip in the steel tub, and then plan to renovate that bathroom with a new enameled steel tub.