I have had an email back from Ichiroya saying he agreed with me re the rayon-ness of this lining and says
From around Taisho era, jinken(rayom) was often used for kimonos and other fabrics, which were silk. Maru obi is often woven with silk and jinken(rayon) blend, so that the lining of vintage uchishiki is rayon is not special case, we think. It might be the original lining ( we are not sure though ).
The Taisho period is 1912 to 1926. My textile expert says that rayon was highly regarded in the 1930s which would fit.