This is a follow up and somewhat of a variant of the question I asked a couple of days ago (see Invariance of set distances with $\varepsilon$-neighbourhoods), and after devoting some research and time in deep thought, I couldn't come up with a solution. For convenience, I'll describe the situation again.
Let $\Omega$ be a non-empty open subset of $\mathbb{R}^n$ and let $K \subset \Omega$ be compact. For any set $A \subset \mathbb{R}^n$ and multiradius $r = (r_1, \dots, r_n) > 0$, define $$A_\text{cube}(r) = \bigcup_{a \in A} \prod_{i = 1}^n \{x_i \in \mathbb{R} : |x_i - a_i| < r_i\}.$$ Suppose that the closure $\overline{K_\text{cube}(r)}$ in $\mathbb{R}^n$ is contained in $\Omega$ for some multiradius $r > 0$. Is there a formula for $$\text{dist}\left(K_\text{cube}(r), \partial\Omega\right)$$ similar to the formula for the usual $\epsilon$-neighbourhood with balls as discussed in my linked question above? In other words, I'm looking for a formula for $\text{dist}\left(K_\text{cube}(r), \partial\Omega\right)$ that explicitly involves $\text{dist}(K, \partial\Omega)$. Of course, here $\text{dist}(A, B) = \inf\{|a - b| : a \in A, b \in B\}$. This is a bit different to the original question I asked because in a sense, distances here are not as "uniform" as for $\epsilon$-neighbourhoods and it doesn't look like you can just "subtract" $r$.
So after some looking around and research, I've come across the following construction. Fix $a \in \Omega$ and let $C(b, \rho) = \prod_{i = 1}^n \{x_i \in \mathbb{R} : |x_i - b_i| < \rho_i\}$ be the cube neighbourhood of $b \in \mathbb{R}^n$ of multiradius $\rho > 0$. For a given multiradius $\rho > 0$, define $$\text{dist}^{(\rho)}(a, \partial\Omega) = \sup\{\lambda > 0 : a + \lambda C(0, \rho) \subset \Omega\}.$$ In other words, the distance from $a$ to $\partial\Omega$ with respect to a multiradius $\rho$ is just the biggest scaling so that the cube $\lambda C(a, \rho)$ stays in $\Omega$. Then it is a fact that $$\text{dist}(a, \partial\Omega) = \inf\left\{\text{dist}^{(\rho)}(a, \partial\Omega) : \rho > 0, \sum_{i = 1}^n \rho_i^2 = 1\right\}.$$ I've tried doing something with this with similar arguments in the linked question, but I couldn't come up with anything fruitful regarding my question. Any suggestions would be greatly appreciated!
