[I want to think that this question has an answer, but it may be more a "community wiki" discussion. Feel free to re-tag.]
What are trig classes like within a universe that's "noticeably"[*] hyperbolic?
Using an appropriate model to peek into such a universe from our Euclidean one shows us that all the fundamental trig relations (Law of Sines, Law of Cosines, etc) involve transcendental functions in both angle measures and lengths. But, then, in our earliest mathematical days, similar triangles allowed us to build a theory of (circular) trig functions based on ratios of lengths of sides, and eventually our mathematics became sophisticated enough to include hyperbolic functions and to be comfortable enough with them and how they might apply to our non-Euclidean models. (We also had the benefit of being able to interpret products as areas of rectangles whose side-lengths correspond to factors.) What if you don't --can't-- draw from our experience base?
How do you even get started developing trigonometry (or explaining it to your hyperbolic trig students[**]) without a concept of similar triangles?
Obviously(?), the Unit Circle is out.
It makes some sense that the Angle of Parallelism would be the fundamental bridge between angle-information and distance-information; one can imagine that hyperbolic people would be "aware" of the phenomenon on some level, and we know that it's a "universal" property. Even so, the most-concise representations of the AoP relationship are transcendental in both angle measure and length. How insightful would a hyperbolic mathematician have to be to discern the equations from tables of observed measurements? And is there a clear path from those equations to, say, (what we know as) the Law of Sines and the Laws of Cosines?
Or perhaps the fundamental figure in an "intrinsic" hyperbolic trig class is the Equilateral Triangle. This idea actually exploits non-similarity to set up a bridge between angle-information and distance-information (with area-information thrown in as a bonus); and it seems that it might be more likely to provide a path to the Laws of Sines and Cosines, since it already relates angles and sides of triangles. But, is it really a particularly helpful starting point? Can you get from there to the Angle of Parallelism relation?
[*] For instance, anyone can pull out a protractor and easily see that triangles have an angle-sum less than 180 degrees.
[**] "hyperbolic" modifying both "trig" and "students". :)