What is the rule behind 一带一路? I know that 一带一路 sounds exactly right in Mandarin for Belt and Road.  But is there a general grammatical rule for why this uses 一?     

It really does not mean "one belt, one road," and if you wanted to say that you would use classifiers for 带 and 路.
It is a title of a project, not a sentence. 

「一带一路」」refers to "The Silk Road Economic Belt and the 21st-century Maritime Silk Road"

「一带」(the one belt) refers to 「丝绸之路经济带」

「一路」(the one road) refers to 「21世纪海上丝绸之路」

Both are specific object, therefore the title cannot be just 「带和路」(belts and roads)

Using 「一个地带和一条通路」would be too wordy for a title. Not to mention 「丝绸之路经济带和21世纪海上丝绸之路」