Have you looked at Dover Saddlery? https://www.doversaddlery.com/saddle...q100000012896/
They advertise cotton pads with cotton lining in many styles.
This particular one reads as all cotton...realize that this does not look to have any padding between layers, just top and bottom layer stitched together with diamond box-stitches to keep shape and reduce bunches made. https://www.doversaddlery.com/quilte...04104148860140
If you are not looking for a padded saddle pad, just a thin layered one then baby pads of 100% cotton you can find for saddle protection, called the underlayment pad.
At one time you could find cotton saddle pads with cotton batting in the middle and cotton flannel on the underside all diamond stitch patterned together so no shift, bunch or wrinkle did your horse suffer from...not sure if they still exist today.
Once you figure out the terminology for what you seek, there are many available in many price ranges.
You do realize that if you are handy with a sewing machine you can purchase mattress covers in 100% cotton in all sizes that you then can cut and shape any way you want, all in white though
100% cotton mattress pads are thick with cushiony batting in the middle, but all material is cotton, no polyester! They're squishy to sleep on.
Cotton mattress pads you can buy online or I know I buy mine in TJMaxx stores cheap.
Yes, I made saddle pads...used a saddle pad shape and size I loved by tracing its shape to the cover leaving a 1/4" extra for stitching together.
I bought some pretty colored wide binding tape and some nylon wide webbing tape to make the girth hold straps and the straps that go to the billets to keep pad in position better. This I accomplished by doing a zig-zag and outline stitch to secure well to the pad.
Being handy with a sewing machine has some good payoffs at times.
My aunt has a embroidery machine that she did my initials on a square pad for me, looked great.
Just remember you need to do embellishments to the top layer of the pad first, then put it together and stitch it.
The saddle pad straps can also be done with Velcro so depending upon saddle slack can be given or removed for neater appearance (never tried that myself but see it advertised on some pads)
I know today in certain show classes a square pad is permitted/encouraged where the saddle pad is permitted to hold the competitor number...haven't yet figured how to do the plastic covering pocket yet though...