Plotting Equatorial coordinates to X,Y plane (simulating telescope/camera view)
This isn’t code for a full implementation, but you could do something like this:

Transform your RA, Dec coordinates (which are spherical polar coordinates, just all with the same $r$) to Cartesian $x, y, z$ coordinates.   (The $r$ value you would use here is arbitrary, for example $r = 1$, and should be the same for all stars, not their real distances; you're just projecting them onto a sphere so you can do the transformation.)

Apply a coordinate frame rotation using your field-center RA, Dec coordinates (or perhaps their complements) as the rotation angles, so that the field center direction is the new $z$ axis.

Plot the resulting new $x, y$ coordinates, ignoring $z$.


You might have to do a further rotation of those $x, y$ coordinates to get the plot oriented the way you want, or maybe there’s a clever way to do that with the proper choice of angles in step 2.