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.)

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 $\omega_1^{CK}$.

Here's another example:

Let $\mathcal{O}^*$ be the "computable analogue" of $\mathcal{O}$ (so $\mathcal{O}^*$ is a computable poset which is an end-extension of $\mathcal{O}$; see Sacks' book for details). Fix an increasing sequence $(\alpha_i)_{i\in\omega}$ of computable ordinals cofinal in $\omega_1^{CK}$ and an enumeration $(p_i)_{i\in\omega}$ of $\mathcal{O}^*\setminus\mathcal{O}$. We can iteratively build a sequence of notations $n_i$ for $\alpha_i$ such that $n_i<_\mathcal{O}n_{i+1}$ and $n_i\not<_{\mathcal{O}^*}p_i$; the downwards closure of $\{n_i:i\in\omega\}$ then is a maximal chain in $\mathcal{O}^*$ and clearly has ordertype $\omega_1^{CK}$. The key point is that by picking a cofinal $\omega$-sequence in $\omega_1^{CK}$ we avoid having to deal with limit stages in this construction, which would otherwise be a problem (we might become maximal too early).

(This is closely related to the construction above, which is unsurprising.)