speaker cable that would work under a rug

I like Pine Tree Audio, they made me a custom braided phono cable, good price.

https://sites.google.com/view/pine-tree-audio/speaker-cables/speaker-cables/rainbow-ribbon

Just google flat speaker wire, Crutchfield, Monoprice and a few others are available. I put the flat braided speaker wire down before I had the wood flooring installed.

Don't forget Silversmith you All.  VERY thin/flat.

Some of the Morrow Audio speaker cables are flat.  I replaced lampwire-type speaker cables in a bedroom mid-fi system with low-end Morrow Audio cables, and they were an improvement.

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.