Why not get some of those rubber puzzle mat pieces? Thicker than cardboard so it should be more insulated and less stiff
https://www.harborfreight.com/4-pie...jIpnJgWBpytPGhY_O1EjZDD6HXQTq2XRoCVXEQAvD_BwE
https://www.harborfreight.com/4-pie...jIpnJgWBpytPGhY_O1EjZDD6HXQTq2XRoCVXEQAvD_BwE