If $\mathcal{C}$ and $\mathcal{D}$ are Morita equivalent by a pair $({_\mathcal{C}}\mathcal{M}_{\mathcal{D}}, {_\mathcal{D}}\mathcal{N}_{\mathcal{C}})$ of bimodules, then there is a natural map of fusion bimodules ${_D}N \otimes_{C} M_{D} \to {_D}D_D$ that preserves Frobenius-Perron dimension (see section 5.1 of Noah's latest preprint with Pinhas Grossman). So the Frobenius-Perron dimensions of elements of $N \otimes_C M$ (which do not depend on knowing $D$) will give Frobenius-Perron dimensions of elements of $D$. Then I think you may be able to use the known possible small Frobenius-Perron dimensions of objects (from your paper with Noah and Frank Calegari) to determine some nontrivial lower bounds on Frobenius-Perron dimensions of simple objects in $\mathcal{D}$, and thus improve on the global dimension bound.
(Or do arbitrarily small numbers of the form $2 \cos (\pi / n)$ already generate the full ring of real cyclotomic integers? Even if so, this method could at least reduce the bound by 1 for weakly integral categories, although that's not a great improvement.)
If $\mathcal{C}$ and $\mathcal{D}$ are Morita equivalent by a pair $({_\mathcal{C}}\mathcal{M}_{\mathcal{D}}, {_\mathcal{D}}\mathcal{N}_{\mathcal{C}})$ of bimodules, then there is a natural map of fusion bimodules ${_D}N \otimes_{C} M_{D} \to {_D}D_D$ that preserves Frobenius-Perron dimension (see section 5.1 of Noah's latest preprint with Pinhas Grossman). So the Frobenius-Perron dimensions of elements of $N \otimes_C M$ (which do not depend on knowing $D$) will give Frobenius-Perron dimensions of elements of $D$. Then I think you may be able to use the known possible small Frobenius-Perron dimensions of objects (from your paper with Noah and Frank Calegari) to determine some nontrivial lower bounds on Frobenius-Perron dimensions of simple objects in $\mathcal{D}$, and thus improve on the global dimension bound.
(Or do arbitrarily small numbers of the form $2 \cos (\pi / n)$ already generate the full ring of real cyclotomic integers? Even if so, this method could at least reduce the bound by 1 for integral categories, although that's not a great improvement.)