A Dm7 and an F chord share some of the same notes. The voicing I was looking for was a descending bass line from the Dm to the Dm7, hence the choice of the 2010 Dm7 voicing. So yes, it looks like an F, but in context, it acts as a Dm7.
We've only got 4 strings, man. Voicing and melody must outweigh everything else.