How do you identify the cross ratio equation of projective geometry from the hyperbolic geometry of relativity? Specifically, what relativistic variables would correspond to A,B,C,D in the standard cross ratio? Does this synchronize with the conformal factor in spacetime?
If by “cross ratio equation” you mean the equation
or something similar, then usually $A,B,C,D$ are assumed to be collinear points in the underlying projective space. According to this post of mine, such a point in the projective space (e.g. on the hyperboloid model for hyperbolic geometry) corresponds to a line through the origin in the space-time diagram of relativistic geometry. So each of $A,B,C,D$ would correspond to the world lines passing through the here-and-now point. Which means that they are essentially equivalent to velocity vectors in space. But the lines have to be coplanar, which means the speed vectors have to point in the same (or opposite) directions, so you can simply assume you're talking about oriented speeds for movement along a straight line. Which doesn't neccessarily mean that it's OK to plug speeds, measured in $m/s$, into the above equation; one would first have to verify that the standard basis used for speed measurements is compatible with this.
Hope this helps. I know my background on projective and hyperbolic geometry is better than my relativistic background, so e.g. I'd have to research whether there are any specific cross ratio formulas established there, and what that conformal factor you're talking about is. If you think my answer doesn't help, let me know in a comment and I may delete it.