This turned out quite a bit better than I expected.
I used a corrugated Split K-FrtBar/Purled Post stitch combo for the in-the-round part (brown & aqua yarns).
And I used its no-turn analog for the strap.
After ripping to the transition many times,
I was finally satisfied with the knit stockinette stitch if I knit 2 yarns at a time.
This creates a fabric that’s about the same weight as the tunisian part, but stretchier.