Thanks for the practical insight, Christopher!

A belt-and-thong arrangement is about the only way I can explain this suspension (ca. 1515), assuming it's accurately depicted by the artist.

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as for the katzbalger suspension, have you considered tablet-woven bands?

i think it makes sense with the silk sash, and you're right about the thickness of tablet weaving. However, I have seen very thin reconstructions made from silk, form the viking period (which is about 700-1066 here in Denmark). Hhey would behave almost as a sash if wide enough.
But, since the depictions show no pattern in the suspension-material, it probably is just one thin long piece of silk fabric.

Two more images, the first is by Niklaus Manuel Deutsch and gives a view of the knotwork


The second is a photo of a modern reconstruction using cord


With regard to the use of fabric I've made decent use of linen when wearing a borrowed katzbalger which had a scabbard but no suspension. Fastend with knots and wound around the scabbard it allowed me to carry the katzbalger in something very close to the 'horizontal' fashion seen in images. Particularly when tied tightly to the breastplate it stayed in place and I had little problem with drawing it when needed.