Ok so here is an answer for the first part of the question, namely how to get an internal CBA-structure from an internal CABA-structure. To get that, one basically has to look at the monadicity proof of $\mathbf{BA} \to \mathbf{Set}$ and make it as explicit and direct as possible.
Notice that for $f : X \to Y$ the induced map $\tilde{f} : P(P(X)) \to P(P(Y))$ is $\tilde{f}(S) = \{A \in P(Y) : f^*(A) \in S\}$.
Assume $B \in \mathcal{C}$ and we are given natural maps $\alpha_X : P(P(X)) \to \hom(B^X,B)$.
If $X$ is an empty set, then $P(P(X)) = \{\emptyset,\{X\}\}$, so we define $0 := \alpha_{\emptyset}(\emptyset) \in \hom(B^0,B)$ and $1 := \alpha_{\emptyset}(\{X\}) \in \hom(B^0,B)$.
If $X$ is a set with two elements, say $X=\{u,v\}$, then $\eta(u) \vee \eta(v) \in P(P(X))$ is actually $\{\{u\},\{v\},\{u,v\}\}$, and its image under $\alpha$ is an operation $\vee : B^2 \to B$.
More generally, If $X$ is any set, then $\bigvee_{x \in X} \eta(x) \in P(P(X))$ is $\{A \in P(X) : A \neq \emptyset\}$, and its image is an operation $\bigvee : B^X \to B$.
Likewise, if $X$ is any set, then $\bigwedge_{x \in X} \eta(x) \in P(P(X))$ is $\{X\}$, and its image is an operation $\bigwedge : B^X \to B$.
If $X$ is a set with just one element, say $u$, then $\neg \eta(u) \in P(P(X))$ is $\{\emptyset\}$, and $\alpha$ maps this to an operation $\neg : B \to B$.
If $\alpha$ is a monad morphism, then $(B,\bigvee,\bigwedge,\neg)$ is an internal complete boolean algebra.
What is still kind of mysterious to me is that apparently we get more structure, or at least properties.