What would the density profile of this shock wave look like, roughly?
The shock observed is effectively the same as the one over a wing. Compression waves are normally considered of infinitesimally small thickness, although due to the non-continuous nature of gases made of particles, this normally makes the shock on the same order of magnitude as the mean free path of a gas molecule (See here for the values at different altitudes). 

For visualization's sake, here's a CFD from a transonic NACA0012 airfoil with a shock on the suction side, and a plot of the density gradient on its surfaces. The step in density is not a pure discontinuity, as it would be physically, due to the relatively coarse mesh used, but I hope it still delivers the idea.