Paul, et al. Thanks again for the comments and tips. Keep 'em coming. Very helpful as I continue to learn.
We were using a Sony ECM-673 (shotgun) on a boom and, as suggested, getting as close to the talent as shots permitted. The problems we had were with the settings on and between the 5D3 and the Beachtek. The tips here will undoubtedly help fix those issues.
Since the 5D3 has manual audio control, I'm not going to address AGC in the below.
Not disregarding the suggestions for mic placement, etc... Just focusing on settings, here is my understanding of the suggestions for settings so far (I'm even listing the rather simple ones, just to make sure - I know, this all looks like common sense to many here).
- Connect the Beachtek to the 5D3 via the Beachtek's line out (not the headphone out)
- Monitor using good headphones from the 5D3 headphone output, with headphone level set high
- Set the Beachtek to stereo mode (as channels can be isolated in post to be made mono or panned, if desired)
- Set the Beachtek mic levels as hot as possible (unless there is clipping) (if there were just an "11"!

)
- Set the 5D3 pre-amp audio levels as low as possible to get a good signal at about -12db (or perhaps even -18db with gain added in post)
- If 5D3 is at its lowest setting and the signal is still too hot (i.e., clipping is likely), back off on the Beachtek mic levels
I agree that the internal recording sounds very good with the proper signal and settings... Good enough not to need to bother with an external recorder. I'm getting there!
Would the NTG-2 be a worthwhile step up from the Sony? The Sony uses phantom power (supplied by the Beachtek, of course). Does the use of phantom power usually result in less signal? In other words, is it better to have a mic that has its own power supply, as a general rule?