is it possible to resolve a constraint connecting arbitrary points on two rigid bodies, in the same manner as an non-rotational particle? That is, by correcting the position, as opposed to applying forces? there are several implementations around using forces/impulses, but we're using verlet integration and for simple constraints the position resolve works so well.

Yes. Constraint forces may have more of a physical basis in reality and provide more realistic results, but explicitly enforcing constraints with verlet integration works well enough for a lot of things. This is the method described in this article, which discusses the physics used for ragdolls in the game Hitman.

Here's another paper you might want to take a look at.
http://web.archive.org/web/20051025222048/http://www.gdconf.com/archives/2004/baltman_rick.pdf