Wow what a great looking wood stove, I am a professional painter by trade so I think I can help you. There is a product that has been out for several years now, and I have used it with great results. This product enables you to remove all rust so it never ever comes back and it is as easy as brushing it on. Just brush it on, it chemically changes the rust and turns the piece to black again, then you just paint over that, the rust will never return again and it is safe for all applications. this product comes in many names so check with your local paint store, they will have it in stock and it is very inepensive and you don't need much product or time.
Now to the paint I think BBQ paint is not good enough and will not last very long, I would suggest car engine block paint, you can buy it at any auto parts store in many colors, in a spray can. This paint has a higher tempiture threshold, and you can get it in several colors and it will last longer without discoloring.
Like i said I have used this before mainly because it is so fast and easy and last forever, trust me this is the way to go, this is how i earn my living this has served me very well, I hope know it will work for you.
AL