If Beer's Law appears to apply to a high concentration (>0.01M) of a solute, is it valid to use for concentration calculation?
Internet consensus seems to define "high concentration" for Beer's Law
  as >0.01M


Keep in mind that Beer's law is an approximation. Look at the more advanced version. Beer's law-advanced version. The statement that a concentration of 0.01 M is the upper limit of Beer's law is incorrect due to several reasons. It is the absorbance value which we should worry about rather than the concentration. One can have a solution which is 0.01 M yet its extinction coefficient is not that large at that particular wavelength, we would still be fine.

This trick is used in atomic absorption spectroscopy, where you choose wavelength, which is not a resonance line of that particular atom. As a result, one can use relatively high concentration without getting a non-linear curve.

Secondly, chemists tend to infatuate little bit with linear calibrations. It is okay to have a non-linear curve. Quadratic equations can be used to fit the empirical curve and one can get accurate analytical results.