speaker cable that would work under a rug

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. 

I did consider doing this but decided against it. Retail on my Morrow Audio speaker cables is $3000+. I don't want to risk damaging them. If that doesn't bother you, Morrow recently redesigned their cables and now they're flatter.