There are categorified analogs of the Chern character, but I don't think of them in the way you're proposing. More precisely, you can take an object in the derived category and assign to it a class in cohomology, and this map factors through K-theory, so the two constructions you're discussing seem to me to be the same.

One way to think of the Chern character is the following. Given any associative, dg or $A_\infty$ algebra, you can define its Hochschild homology. This is the recipient for a universal trace map from the algebra, and more generally for any "finite" module (perfect complex) you get a class (its character) in Hochschild homology. Given more generally a (dg or $A_\infty$) category you can similarly define its Hochschild homology and a character map for "finite" objects (which factors through the K-theory of the category), which agrees with the above when your category is modules over an algebra (which it usually is, noncanonically).

To "categorify" you can replace an algebra by an associative algebra object in any symmetric monoidal $\infty$-category. Its Hochschild homology is defined as an object of said category and again there's a Chern character map for "finite" modules. Why is this a categorification? for example you can take your associative algebra to be some derived category of sheaves with a monoidal structure (eg coherent sheaves or $\mathcal{D}$-modules or.. with tensor product or some convolution product), and then its Hochschild homology is itself a category. Thus module categories will have Chern characters which are
objects of this homology category. This is (one way to think of) the notion of a
"character sheaf" in representation theory (where our associative algebra is sheaves on a group with convolution, and module categories are categories with a nice action of the group, and their Chern character are adjoint-equivariant sheaves on the group, ie categorified class functions).

(This story is by the way a special case of the Cobordism Hypothesis with Singularities of Jacob Lurie -- in fact just of its one-dimensional case.. our algebra objects are assigned to a point, their Hochschild homology is assigned to the circle, modules are allowable "singularities" in the theory and their Chern character is attached to a circle with a marked "singular point")