1. Is it correct to reply with either A or B in the exchange above?-- Only B sounds natural.
2. If yes, how are they different in meaning?
3. Can I also reply with "That is all"? -- It is better than A, but still not quite natural. If it were a friendlier dialog, 'That's all, thanks' would sound good.
I see B is the best answer and "That is all" is still not quite natural. Is it not quite natural because it should be hyphenated to make it sound natural or because it should add "thanks" to sound natural?