I think the following is true, and what I was looking for.

A tensor functor $F: \mathcal{C} \to \mathcal{D}$ is called dominant (sometimes called "surjective") if for any $Y: \mathcal{D}$, there is an $X: \mathcal{C}$ such that $Y$ is a subobject of $FX$.

Theorem: Any tensor functor of fusion categories factors as a dominant functor and a full inclusion.

Proof: First, define the category $\operatorname{Im}F$. Its objects are all subobjects in $\mathcal{D}$ of all $FX$, where $X: \mathcal{C}$, and all other objects isomorphic to them. The morphisms of $\operatorname{Im}F$ are such that it is a full subcategory of $\mathcal{D}$.

By construction, $F$ restricted to $\operatorname{Im}F$ is dominant. Also $\operatorname{Im}F$ is a full subcategory, so it inherits all additional structure like the pivotal/spherical structure, a braiding or a ribbon structure.

I think the following is true, and what I was looking for.

A tensor functor $F: \mathcal{C} \to \mathcal{D}$ is called dominant (sometimes called "surjective") if for any $Y: \mathcal{D}$, there is an $X: \mathcal{C}$ such that $Y$ is a subobject of $FX$.

Theorem: Any tensor functor of fusion categories factors as a dominant functor and a full inclusion.

Proof: First, define the category $\operatorname{Im}F$. Its objects are all objects of $\mathcal{D}$ that are isomorphic to a subobject of an $FX$, where $X$ is any object of $\mathcal{C}$. The morphisms of $\operatorname{Im}F$ are such that it is a full subcategory of $\mathcal{D}$.

By construction, $F$ restricted to $\operatorname{Im}F$ is dominant. Also $\operatorname{Im}F$ is a full subcategory, so it inherits all additional structure like the pivotal/spherical structure, a braiding or a ribbon structure.