Alas, in my opinion bamboo IS rayon (fabric made by running
wood pulp through a chemical process). The natural alternative is silk. R wanted a drapier fabric for her blouse and rayon will do that without costing a fortune.
Linen can be soft after a few washings, but it tends to be expensive. R, I think, grows and weaves her own linen so she knows its ways.
Wool crepe might work, but again, expensive and not easy care.
Quilting cotton is cheap, easy care, and a bit like wearing typing paper. If there are softer cottons out there, I'd like to know about them.