Crepuscular rays rotate?
Nope.
Anti-crepuscular rays are just the opposite ends of the crepuscular rays.
They are the same rays.
They don't rotate, otherwise it would mean that "the Dome" rotates in north-south direction with east-west axis.
Also, northern stars don't extend into southern stars.
We have unique set of the stars at the north and completely different set of the stars at the south.
Since they rotate it means that the dome rotates in east-west direction.
If crepuscular / anti-crepuscular rays rotate AND northern and southern stars rotate,
it would mean that "the Dome" rotates in multiple directions simultaneously.
Now, back to the OP:
The whole story about rotation of stars doesn't explain at all how is Sigma Octatntis
at the same moment visible due south from South Africa and from Argentina
before we even take time to observe any further star rotation.