How do most fur buyers measure beaver, across the shoulders or north and south east and west?

From what I've seen most measure north-south, east west. If you leave the nose on most will place the yardstick just below the nose.

At one point in time, beaver were measured in an “X” pattern across the leg holes. Not sure when that went out of style.

When the buyer would measure length plus width, a few trappers would use just 4 nails and stretch out the length and width as much as possible. They were a Diamond shape rather than oval/round. They were called “Michigan diamonds”. Kind of a fun shape.

A fur buyer told me years ago that he measured north/south + east/west unless it was a "messed-up shape". Then he would measure across the leg holes

Yup. I've seen that. Whichever system that makes for a smaller size.