The main problem with dual algebras for non-separable C*-algebras is that they need not be functorial. Given representations $\rho_A \colon A \to \mathcal{B}(H_A)$ and $\rho_B \colon B \to \mathcal{B}(H_B)$ and a $*$-homomorphism $\phi \colon A \to B$, one wants to define the induced map $\phi_* \colon \mathcal{D}(B,\rho_B) \to \mathcal{D}(A, \rho_A)$ by $\phi_*(T) = VTV^*$ where $V \colon H_B \to H_A$ is an isometry such that $V^* \rho_A(a) V$ is equal to $\rho_B(\phi(a))$ modulo compact operators. Such isometries are constructed using Voiculescu's theorem, which requires that $A$ is separable. Without functoriality the whole theory pretty much collapses; for instance, one proves that K-homoogy is independent of the ample representation used to define the dual algebra by arguing that the K-theory map induced by $\phi_* \colon \mathcal{D}(B, \rho_B) \to \mathcal{D}(A, \rho_A)$ is independent of the isometry $V$.

For Kasparov's model of K-homology, you don't run into this problem: if $(\rho, H, F)$ is a Fredholm module over $B$ then $(\rho \circ \phi, H, F)$ is a Fredholm module over $A$. Still, there are problems: without Voiculescu's theorem you don't have the excision theorem in either model (at least the standard proof doesn't work) and without excision you don't have long exact sequences.

None of this should be too much of a concern, however - most C-algebras for which K-theory is useful are separable (e.g. $C(X)$ for $X$ compact, crossed products of $C(X)$ by a locally compact group action). The main source of non-separable C-algebras is the theory of von Neumann algebras, and K-theory/homology doesn't tell you anything about von Neumann algebras anyway.

The main problem with dual algebras for non-separable C*-algebras is that they need not be functorial. Given representations $\rho_A \colon A \to \mathcal{B}(H_A)$ and $\rho_B \colon B \to \mathcal{B}(H_B)$ and a $*$-homomorphism $\phi \colon A \to B$, one wants to define the induced map $\phi_* \colon \mathcal{D}(B,\rho_B) \to \mathcal{D}(A, \rho_A)$ by $\phi_*(T) = VTV^*$ where $V \colon H_B \to H_A$ is an isometry such that $V^* \rho_A(a) V$ is equal to $\rho_B(\phi(a))$ modulo compact operators. Such isometries are constructed using Voiculescu's theorem, which requires that $A$ is separable. Without functoriality the whole theory pretty much collapses; for instance, one proves that K-homology is independent of the ample representation used to define the dual algebra by arguing that the K-theory map induced by $\phi_* \colon \mathcal{D}(B, \rho_B) \to \mathcal{D}(A, \rho_A)$ is independent of the isometry $V$.

For Kasparov's model of K-homology, you don't run into this problem: if $(\rho, H, F)$ is a Fredholm module over $B$ then $(\rho \circ \phi, H, F)$ is a Fredholm module over $A$. Still, there are problems: without Voiculescu's theorem you don't have the excision theorem in either model (at least the standard proof doesn't work) and without excision you don't have long exact sequences.

None of this should be too much of a concern, however - most C-algebras for which K-theory is useful are separable (e.g. $C(X)$ for $X$ compact, crossed products of $C(X)$ by a locally compact group action). The main source of non-separable C-algebras is the theory of von Neumann algebras, and K-theory/homology doesn't tell you anything about von Neumann algebras anyway.