I was in Costa Rica 6 months ago, and now that you mention it I don't think we saw any rugs ... everything was tile or wood ...
Your making this difficult
... Guess, there's no perfect solution ...
My next thought would be to find a way to run the wires and cover them over (like using the thin wire along the wall/in-wall and under an entended threshold if there is a door) ... then putting a mat over it as it crosses the tile to get under the couch ... (personally I'd just put low voltage and power boxes in the floor (but that's me).
Guess it's easiest to skip the whole thing ...
