speaker cable that would work under a rug

I strongly recommend not running your speaker cables under a rug, especially if it is a high traffic area.  There is a potential for dielectric damage causing a short that will damage your amplifier and speakers, and result in a fire hazard if your amplifier is damaged in a manner that it will overheat.  A fire hazard directly from the shorted cable itself is much more remote due to low voltage and current, but you still can fry your speakers and amp.   If you choose to do so make sure you use cable raceway of adequate dimensions to protect the speaker cables.  

None ……just wouldn’t do it . 

Nordost Flatline or Leif3 Purple Flare are very flat and use hard FPE insulation that is very durable.  If you run it between a carpet pad and area rug, it will last you indefinitely. Wall to Wall would be more challenging. 

Typically, the maximum spread of the speaker listener triangle is 7’ or so and you want to run under the carpet? If true, doesn’t sound like you will be stepping on it... so most flat audio cables will work.