How to plot curvilinear coordinates in the frame of a Cartesian telescope field of view
Most CCD or photographic images are equivalent to a gnomonic projection.
I suppose this is true of the Kepler CCDs; while the Schmidt camera focus surface is curved, the CCDs have individual field flatteners.
If the projection is not exactly gnomonic, the difference is not very great in a small field.

As you found, a single gnomonic projection doesn't work so well for areas much wider than 90&deg;.
TheSky user guide
says it uses orthographic projection for fields smaller than 60&deg;.
The slightly curved RA lines and slightly oblique intersections near the corners of the chart in question are consistent with that.
Other projection choices are also valid, e.g. stereographic projection is common in maps showing a hemisphere at once.

In any case you'd need to transform the individual image tiles to the same projection as the composite overview.
This will change the tiles' shapes or sizes, but the relation between map coordinates and sky coordinates will be consistent.