If I understand correctly, $\chi(H^\partial)$ is the least number of colours which suffice to colour the edges of $H$ so that each vertex is incident with edges of at least two different colours. And it seems to me that $H$ is isomorphic to $(H^\partial)^\partial$ provided that, for any vertices $x,y\in V(H)$, there is an edge $e\in E(H)$ such that $|e\cap\{x,y\}|=1$. I believe the answer is affirmative for all $\kappa,\lambda\ge2$. Each of the example hypergraphs is either an ordinary graph, or the dual of a graph, or the disjoint union of a graph and the dual of a graph.
If $\kappa\ge4$ and $\lambda=2$, let $H=K_\kappa$ (the complete graph of order $\kappa$).
If $\kappa=2$ and $\lambda\ge4$, let $H=K_\lambda^\partial$.
If $\kappa\ge4$ and $\lambda\ge4$, let $H=K_\kappa\cup K_\lambda^\partial$ (disjoint union).
If $\kappa=2$ and $\lambda=2$, let $H=C_4$.
If $\kappa=3$ and $\lambda=3$, let $H=K_3$.
If $\kappa=3$ and $\lambda=2$, let $H=K_4-e$.
If $\kappa=2$ and $\lambda=3$, let $H=(K_4-e)^\partial$.
If $\kappa\ge4$ and $\lambda=3$, let $H=K_\kappa\cup K_3$.
If $\kappa=3$ and $\lambda\ge4$, let $H=K_3\cup K_\lambda^\partial$.
Theorem. For any cardinals $\alpha,\beta\ge2$ there is an $\alpha$-uniform hypergraph $H$ with $\chi(H)=\beta$ and $\chi(H^\partial)=2$.
Proof. Let $V=\bigcup_{\xi\in\beta}V_\xi$ where the sets $V_\xi$ are pairwise disjoint and $|V_\xi|\gt\alpha\beta$. Let $E=\{e\in[V]^\alpha:|\{\xi\in\beta:e\cap V_\xi\ne\varnothing\}|\ge2\}$.
Plainly $H=(V,E)$ is an $\alpha$-uniform hypergraph and $\chi(H)=\beta$. To see that $\chi(H^\partial)=2$ color each vertex red or blue so that each set $V_\xi$ contains at least $\alpha$ vertices of each color. Then each vertex of $H$ is contained in both a monochromatic edge and a bichromatic edge, whence $\chi(H^\partial)=2$.
Best Answer
Fred Galvin had conjectured that the answer is "yes" for graphs in [1] (conjecture 2), in his paper he showed that the variation of the problem to induced graphs is consistently false: Assume $2^{\aleph_0}=2^{\aleph_1}<2^{\aleph_2}$ there exists a graph $(V,E)$ with a chromatic number of $\aleph_2$ but for no subset $V'\subseteq V$ we have that the chromatic number of $(V', E\cap [V']^2)$ is $\aleph_1$.
Later in [2] Péter Komjáth had shown the consistency of the failure of the above conjecture: he gave a model in which $2^{\aleph_0}=2^{\aleph_2}=\aleph_3$ where there exists a graph with chromatic number of $\aleph_2$ but no subgraph with chromatic number $\aleph_1$.
Lastly, in [3] Shelah showed a partial positive direction: assume $V=L$, then for any $(V,E)$ and $\kappa$ such that $|V|<\kappa^{+\kappa}$ and $\chi(V,E)>\kappa$, then there exists a subgraph of chromatic number exactly $\kappa$. Further more, he had showed that under $V=L$ there is no counterexample of cardinality $\aleph_2$: every graph with cardinality $\aleph_2$ contains a subgraph in any smaller chromatic number.