Map a texture onto a hyperbolic triangle
My guess is that to get the texture to look right, you'll have to subdivide the interior of the triangle as well, and approximate the non-linear UV mapping within it.

Currently, it looks like you're subdividing around the edges of the triangle, and forming a fan of smaller triangles between the edge and the incenter. This would be fine if you were just rendering the triangle in a solid color. But when applying a texture, you're getting discontinuities at the boundaries of those subdivisions, because the texture is just stretched linearly across each subdivided triangle and not correctly approximating the hyperbolic mapping along the radial direction.

You'll need to subdivide along both axes, something like this:



and position all the vertices appropriately in screen space and UV space to approximate the hyperbolic coordinates in the interior of the triangle. If you subdivide finely enough, this should produce the illusion of a continuously curving texture mapping.