You seem to be using a strange definition of "measurement" that requires observers to be intentionally naive. Why in the world would anyone try to measure the angle between AB and AC with a moving protractor?
All geometric quantities
are invariant under coordinate transformations. In Euclidean space, geometric quantities include angles and distances. An angle is always measured between two lines at the point they intersect. A distance is always measured between two points along the line that connects them. In Minkowski space, geometric quantities include angles, distances, and relative velocities. Relative velocity is really just the "angle" between two worldlines.
I've used the term "relative velocity", but you should note that ALL geometric quantities are already
"relative". An angle is always an angle between two lines. One cannot say "The angle of line AB is 30 degrees", that makes no sense. Likewise, a distance is always a distance between two points.
You realize that your inability to answer this question unambiguously proves
that Dale is in fact not using a coordinate system?
You'll have to explain. A continuous map from what space into what space?
This statement makes no sense. You don't seem to be using the words "vector" and "mapping" correctly. Furthermore, I have not once made any mention of vectors, so it's irrelevant anyway.