The best revision of the wordy sentence is A) The brown dog sitting under the table belongs to the next-door neighbor.
Option A) is the correct one because the phrase "that is brown", which is postmodifying the noun, has been eliminated and "brown" has become a premodifier of "dog" instead. Therefore, this change makes possible to express the same idea using less words. Moreover, the conjunction "and" and the defining relative clause "who happens to live next door" have also been deleted. The adjective "next-door", which is modifying "neighbour", is replacing the defining relative clause and, at the same time, making the sentence less wordy.
