EDIT1:

In what follows I am prepending some considerations regarding intersection of two small circles on a sphere resulting in two diangles, the minor and major diangles or lunes.

Each diangle/lune on either end of two cutting planes has a hinge axis of intersection line to define variable dihedrals. It makes $\alpha,\beta$ inplane of small circles and $gamma$ in the sphere's tangent plane at either end.

The dihedral angle between the two planes is $\delta. $

three dihedrals are taken when considering three geodesic great circle arcs for a spherical triangle of three angles in the tangent plane.

However it is not necessary here in lune/diangle situation. The boundaries need not be geodesics/great circles but can be small circles. Dihedral angle that comes about in the middle $\gamma$

In the next edit I shall extend to a three small circle spherical trigonometry.

The Cosine Rule in Spherical trig is equally valid here with no geodesics, even with small latitude /parallel circles

In the triangular pyramid shown we consider four triangles (two right triangles on a common hinge/fulcrum unit length normal to a striped triangle with a dihedral $\delta$ angle and the outer big yellow triangle containing compound angle $\gamma$).

We derive Cosine Rule in Spherical trigonometry indirectly avoiding representation of sphere radius.

By applying Cosine Rule in striped triangle

$$ c^2= \tan^2\alpha+\tan^2\beta-2\tan\alpha \tan\beta \cos \delta $$

By applying Cosine Rule in larger yellow triangle containing compound angle $\gamma$

$$c^2= \sec^2\alpha+\sec^2\beta-2\sec\alpha \sec\beta \cos \gamma$$

Eliminate $c^2$ to simplify we get Cosine Rule in spherical trigonometry

$$\cos \gamma= \cos\alpha\cos\beta+ \sin \alpha \sin \beta \cos \delta $$

We have used plane trig and embedded a pyramid into $\mathbb R ^3 $ without explicit reference to a sphere:

Now how can we draw the corresponding figure in hyperbolic geometry:

$$\cos \gamma= \cosh\alpha\cosh\beta+ \sinh \alpha \sinh \beta \cos \delta \,? $$

Considering simpler cases visualization... We can draw for right triangle $\delta= \pi/2$ the pyramid but how to at least draw it for hyperbolic geometry representation and result ?

$$ \cos \gamma= \cos\alpha\cos\beta \, \rightarrow \cos \gamma= \cosh\alpha\cosh\beta \,? $$

Thanks in advance for geometric considerations in hyperbolic geometry without explicitly bringing in the pseudosphere.

Regards