This is perhaps a simplification of the example you give: let $H$ be a Harrison order and let $P$ be the poset of pairs $(a,b)\in H$ with $a<b$, ordered by $$(a,b)\le (a',b') \quad\iff\quad a\le a'\mbox{ and }b\ge b'.$$ Note that in a chain in $P$, we do *not* require both coordinates to simultaneously change.

Let $(\alpha_i)_{i\in\omega}$ be cofinal in $\omega_1^{CK}$. Fix a descending sequence $(h_i)_{i\in\omega}$ in the Harrison order such that the ill-founded part of $H$ is exactly the union of the final segments given by the $h_i$s. Then - identifying elements of the well-founded part of $H$ with computable ordinals - the sequence $$(\alpha, h_{\min\{k: \alpha_k>\alpha\}})$$ is a maximal chain in $P$ with ordertype $\omega_1^{CK}$.

(Incidentally, I believe the "computable analogue" $\mathcal{O}^*$ of Kleene's $\mathcal{O}$ also has maximal length-$\omega_1^{CK}$ chains, but I can't immediately recall the proof.)