So, a segment and a circle center is given in the Klein model:
and the task is to construct points, say $D$, for which $CD$ is congruent to $AB$ -- congruent in the hyperbolic sense.
This is how one can construct as many such points, as she wants.
Draw the straight line containing $AB$ and draw a movable straight line through $C$ as shown below:
Here $H$, the "handle" is the point that, if moved, will turn the right straight around the circle center $C$. Note that the part of the white lines within the circle are the hyperbolic straights. Also, in the figure above four yellow lines are drawn. I just hope that I don't have to further explain the construction of the yellow lines.
The next step is to connect the intersection points of the yellow lines with a straight line (red). Then construct the blue line as shown below. Finally, construct the black line that will give point $D$, a point of the circle we try to construct.
Now, move the handle around $C$. Then $D$ will trace the hyperbolic circle.
Note that such a dynamic construction and tracing can be done by a dynamic geometry software like Geogebra or Cinderella...
But, if tracing is not allowed then think of the following fact: five points determine an ellipse. So, it would be enough to construct five $D$'s.