With adequate assumptions ("$\mu$-reach" bounded below) similar to your intuition about possible failure cases, Theorem 4 in:
https://ljk.imag.fr/membres/Boris.Thibert/Publications/RR_stabilityCurvatureMeasure.pdf
will guarantee that the perimeter of the $\epsilon$-neighborhood of $D_n$ will converge to the perimeter of the $\epsilon$-neighborhood of the limit. The speed of convergence is uniform in $\epsilon$. From there, it should be enough (and not too hard assuming uniformly bounded total curvature) to show that the perimeters of $\epsilon$-neighborhoods of $D_n$ converge to the perimeters of $D_n$ as $\epsilon$ go to zero, uniformly in $n$.