With adequate assumptions ("$\mu$-reach" bounded below) similar to your intuition about possible failure cases, Theorem 4 in:

https://geometrica.saclay.inria.fr/team/Fred.Chazal/papers/ccslt-scm-09/ccslt-scm-09.pdf

will guarantee that the perimeter of the $r$-neighborhood of $D_n$ will converge to the perimeter of the $r$-neighborhood of the limit.

Also, it shouldn't be too hard (assuming uniformly bounded total curvature) to show that the perimeters of $r$-neighborhoods of $D_n$ converge to the perimeters of $D_n$ as $r$ go to zero, uniformly in $n$.

It remains to to show that the limits can be "swapped". I believe that choosing $r$ to be a suitable function of the Hausdorff distance will work, thanks to the explicit form of the approximation error in Theorem 4 in the above paper.

EDIT: this answers a different (or part of the) question, as Leo Moos remarked. To get an answer to the question, one would need to show that the $\Omega_n$ converge in the Hausdorff distance, and apply the answer to them instead of $D_n$.

With adequate assumptions ("$\mu$-reach" bounded below) similar to your intuition about possible failure cases, Theorem 4 in:

https://geometrica.saclay.inria.fr/team/Fred.Chazal/papers/ccslt-scm-09/ccslt-scm-09.pdf

will guarantee that the perimeter of the $r$-neighborhood of $D_n$ will converge to the perimeter of the $r$-neighborhood of the limit.

Also, it shouldn't be too hard (assuming uniformly bounded total curvature) to show that the perimeters of $r$-neighborhoods of $D_n$ converge to the perimeters of $D_n$ as $r$ go to zero, uniformly in $n$.

It remains to to show that the limits can be "swapped". I believe that choosing $r$ to be a suitable function of the Hausdorff distance will work, thanks to the explicit form of the approximation error in Theorem 4 in the above paper.

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 $n$.

With adequate assumptions ("$\mu$-reach" bounded below) similar to your intuition about possible failure cases, Theorem 4 in:

https://geometrica.saclay.inria.fr/team/Fred.Chazal/papers/ccslt-scm-09/ccslt-scm-09.pdf

will guarantee that the perimeter of the $r$-neighborhood of $D_n$ will converge to the perimeter of the $r$-neighborhood of the limit.

Also, it shouldn't be too hard (assuming uniformly bounded total curvature) to show that the perimeters of $r$-neighborhoods of $D_n$ converge to the perimeters of $D_n$ as $r$ go to zero, uniformly in $n$.

It remains to to show that the limits can be "swapped". I believe that choosing $r$ to be a suitable function of the Hausdorff distance will work, thanks to the explicit form of the approximation error in Theorem 4 in the above paper.