The difficulty in finding the apparent dip is that one cannot assume the cross section line maintains one trend. Nor is there a built-in OGR function (to my knowledge) that finds the angle(s) of intersection between two lines.
I solved this problem by creating an arbitrary north-south trending line offset a known distance from the determined intersection point. From there it is super easy to find the azimuth of the cross section line! Note that this method will fail on cross section lines oriented exactly north-south.
Next step: making the calculated apparent dips accessible with distances along the cross section line.
Projected Strikes in Magenta, Cross Section Line in Green. Note that rotated strike and dip symbols pivot around the point's geographic coordinate, thus the appearance of offset (the strike lines go right through the point). Buffer was 100 meters. Arbitrary north-south oriented lines aren't in this screen shot. Viewed in QGIS.