Hi Alex,
Have you considered a disk shape to make a Möbius strip?
It seems that making a Möbius ring out of a rectangular ribbon shape generates so many tension lines in different directions and requires a lot of force to tame it—but on the other hand, a disk-shaped ribbon produces a much smoother result with far less tension.
Here's a test on paper: I just cut the disk on the red line and then twisted it to form the Möbius shape.