This is about compass and straightedge. Draw the half-line beginning at A and passing through J, meeting the circle at a point we will call, say, R. Let the distance between A and R be called $r.$ Let the distance between A and J be called $j.$ We need to draw the point W along the ray AJ with distance from A exactly $ \beta = \frac{r^2 + j^2}{2j}.$ There is a standard construction for distance $\frac{r^2}{j},$ called the polar. Draw the line perpendicular to AJ through J. This line intersects the circle twice, pick one, call it T, draw the tangent to the circle at that point T (perpendicular to the radius through T), find the point X where this tangent intersects AJ. This is the point at distance $\frac{r^2}{j}$ from A. Along AJ, draw the point farther out than X by exactly $j,$ call this Y. Y is at distance $ \frac{r^2 + j^2}{j}$ from A. Finally, draw the midpoint Z of the segment AY. The distance between A and Z is $ \beta = \frac{r^2 + j^2}{2j}.$ Let line $\Lambda$ be the perpendicular to AZ through Z. Note that $\Lambda$ is not the polar of J with respect to the circle. Instead, every circle with a center on $\Lambda$ that passes through J will be orthogonal to the circle passing through your C, G, D, F. I do not know a name for this line.
Let line $\Delta$ be the line through C and D. This is called the "chordal." Also called the "power line" or "radical axis." It is, in Dorrie's prose, the "locus of the centers of all circles that are perpendicular to two given circles." In our case, the two given circles intersect, in the points C,D, and we have "The chordal of two circles that intersect is the secant of intersection."
Lines $\Lambda$ and $\Delta$ intersect in a point, call it H. Draw the circular arc with center at H and passing through J. Done.
This material is very old and hard to find. One of the best books giving at least part of this is 100 Great Problems of Elementary Mathematics by Heinrich Dorrie. The trouble these days is that architectural and engineering drafting is done by CAD systems and the like. I grew up playing with compass and straightedge.
Notice that nothing I have mentioned is in any way about hyperbolic geometry. I have told you how to draw a circle through J that meets two other circles orthogonally, the one with center A passing through CGDF, and the other passing through CMD.
It is possible to read about the "chordal," but I made up the other line, so I thought I had better include an example with numbers. Let the original circle be in the $xy$ plane with center A at $(0,0)$ and radius 5, so points CGDF would all satisfy $x^2 + y^2 = 25.$ But then let the point J lie along the $x$-axis at $(1,0).$ In this case the line I am calling $\Lambda$ becomes the vertical line at $x=13.$ Two example centers on $\Lambda:$ first, with center at $(13,0),$ the circle with radius 12 passes through J, and meets $x^2 + y^2 = 25$ at the two points $(\frac{25}{13}, \; \pm \frac{60}{13} ),$ and it is a minor exercise to show that the radii meet orthogonally, that is, the two circles are orthogonal. Second, take the circle with center at $(13,16)$ and radius 20. The circle does pass through J, and meets $x^2 + y^2 = 25$ at the two points $(-3,4)$ and $(\frac{77}{17}, \; \frac{-36}{17} ).$ Same comment about orthogonal circles, easy enough to verify. Indeed, the whole thing, in coordinates, is an exercise in second-year calculus, using two-variable functions and gradients.