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.

(A) Here is how to get an internal CBA-structure from an internal CABA-structure. A reference is Formula 1.5.22 in E. Manes, Algebraic theories.

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.

(B) This is answered by Proposition VII.1.16 in Johnstone, Stone spaces: A complete boolean algebra is atomic iff it is completely distributive, i.e. we have $$\bigvee_{i \in I} \bigwedge_{j \in J} v_{i,j} = \bigwedge_{s:I \to J} \bigvee_{i \in I} v_{i,s(i)}.$$

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, probably just properties.

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.