@cgunn3 nice whitepaper. I also am not an expert in multi-linear algebra, but have been studying differential geometry in parallel with learning GA.
Your geometric notion of a metric free way of getting scalar coordinates from a geometric object is very much like what k-forms are designed for. Often I see the geometric description of how a 1-form measures a 1-vector is by counting the number of 1-form iso-surfaces ( planes ) pierced by the 1-vector . Flip that around to the description you give of displacing basis planes and you have the same effect.