Invertibility threshold for Nevanlinna quotient algebras

Let $\mathcal{N}$ be the Nevanlinna class and let $B$ be a Blaschke product. Consider the natural necessary condition for invertibility of $[f]$ in the quotient algebra $\mathcal{N} / B \mathcal{N}$ : "$|f| \ge e^{-H} $ on the zero set of $B$, for some positive harmonic function $H$". For large enough functions $H$, this is almost a sufficient condition if and only if the function $- \log |B|$ has a harmonic majorant on the set $\{z\in\mathbb{D}:\rho(z,\Lambda)\geq e^{-H(z)}\}$.
We thus study the class of harmonic functions $H$ such that this last condition holds, and give some examples of $B$ where it can be entirely determined.