Who is the carpet manufacturer. Contact them to get recommendations. Wool rugs often have a coarse backing that can catch on a needle punched felt pad and be held in place to avoid stretching and buckling. A natural rubber pad is another option to hold the rig in place on a wood floor. Myers Carpet in Dalton Georgia will have the best product.

I alwaays use a good quality felt pad under most area rugs , I am not sure what is causing this issue but adding a rubber backing does not make sense for a fix. IMO

Have you been able to pull the wrinkle out and it just doesn't stay flat?

thin rugs on top of a carpet creates issues, but the quality of the rug makes all the difference.

I realize it's not the dilemma but

... have you tried moving the bed to face the windows? with the chairs under the horizontal painting?