I would use silk/satiny material. You're right about fur being too warm (in the daytime, anyway).
I've been rubbed raw by backpack straps in the past, because I am overly fond of canvas-strapped military bags worn with tank tops.
If I were going to use my favorite military bag this year, this was the plan: I would pad the straps with neckties. I would've washed the ties once in mild detergent (because ties rarely if ever get washed and tend to carry bacteria, or if new: dye) rinse thoroughly and press, then open the ties down the back and fit them down the straps, and hand sew in place. One could also use a "winding" action; ties are long enough to permit this.
I have a new bag this year and the straps are kinda padded and nowhere near as rough, so I'm probably not going to try it this time.
But I'm going to make damn sure to wash my "under-strap" skin each night with lemon juice, then apply lotion, even Neosporin if I need it. Every night.