I think the answer is a no. If you take the topologists sine curve, together with adjoining the two end points of the sine curve via a different route, then the resulting set is rectifiably connected but has infinite diameter (with respect to the rectifiable curve metric you introduced).
My guess is to look at a closed, bounded subset of R^2 with an inward cusp. Like the closure of the bounded component of a standard cardioid. There can be no bilipschitz map of this set to the same set but with the length metric.
