Generalization: In order to generalize to higher cardinals (up to the first inaccessible), we will need to use $\kappa$-centered ideals on $\kappa$ (instead of $\kappa$-dense). An ideal $I$ is $\kappa$-dense if there are collection of $\kappa$ filters $\mathcal{F}_\alpha$, $\alpha < \kappa$ such that $I^+ = \bigcup_\alpha \mathcal{F}_\alpha$.

By a theorem of Foreman (appears in the Handbook, page 1047, Theorem 7.57), it is consistent relative to a huge cardinal that every regular cardinal $\kappa$ carries a $\kappa$-complete $\kappa$-centered ideal. Moreover, for accessible $\kappa$, those ideals (and also their restriction) are not $\kappa$-saturated, so every positive set $A$ can be partitioned into $\kappa$ positive sets.

Let's assume that in $V$ for every regular $\kappa$, there is $\kappa$-centered $\kappa$-complete non-trivial ideal. Assume also $GCH$, for simplicity.

We will show by induction on cardinals $\lambda$ less than the first inaccessible (if there is one), that the product of $\lambda$ copies of the Cohen forcing collapses $2^{\lambda}$. In fact we will construct the coding method (relatively) explicitly.

Case 1: $\lambda = \mu^+$. This is similar to the $\omega_1$ case above. Let $I$ be a $\lambda$-centered, $\lambda$-complete ideal on $\lambda$ as witnessed by $\{\mathcal{F}_i \mid i < \lambda\}$. We will generically code every element $y$ of $2^\lambda$ by an ordinal from $\lambda$ and a natural number (which in turn are generically coded by a natural number, by the induction hypothesis). Let $p$ be a condition and assume that $n$ is a natural number such that $C = \{\alpha < \lambda \mid n\notin \text{dom }p_\alpha\}\in I^+$. Assume that this set is in $\mathcal{F}_\gamma$. Since the difference between all two sets in $\mathcal{F}_\gamma$ is in $I$, we can pick an arbitrary element there and split it, as before, to $\lambda$ sets - $\{F_i \mid i < \lambda\}$. Again, those sets can be assumed to be disjoint and their union is $C$, up to an element from $I$. We continue, and extend $p$ to code $y$ in the generic in the same way as above: $q\leq p$, $q_\alpha(n) = 0$ iff $\alpha \in F_i \cap C$ and $i\in y$ so $y$ will be coded by $(\gamma, n)$ in the generic filter.

Case 2: $\lambda$ is singular. Let $\{\mu_i\mid i < \text{cf }\lambda\}$ be a cofinal sequence, $\mu_0 > \text{cf }\lambda$. We split the product into a product of $\text{cf } \lambda$ parts. It is clear that every element of $2^\lambda = \prod 2^{\mu_i}$ is coded by an element in $(\omega^{\text{cf }\lambda})^V$, but the latter is coded with a natural number, by the induction hypothesis.

It is not clear at all if we actually need the large cardinals here.

It is not clear at all if we actually need the large cardinals here. Also, it is not clear if it's possible to generalize this method. A direct attempt to generalize this method would require the existence of dense ideals on both $\omega_1$ and $\omega_2$ which is an open problem (see comments below).

Split $C\setminus B$ into $\omega_1$ sets from $A$, up to an error in $I$, $\{F_i \mid i < \omega_1\}$. We can assume that for every $i < j < \omega_1$, $F_i \cap F_j = \emptyset$ (using the $\sigma$-completeness). Code an element of $y \in \mathcal P(\omega_1)$ by coloring the components: $q\leq p$, $q_\alpha (n) = 1$ iff $\alpha \in F_i$ where $i \in y$. Now, every $y \in \mathcal{P}(\omega_1)$ is coded by a real from $V$ which in turn coded by a natural number.

Assuming that there is also an $\omega_2$-complete $\omega_2$-dense ideal on $\omega_2$ we can continue to $\lambda = \omega_2$: code a real, use it to code subset of $\omega_1$ which translate to an element in the dense subset corresponding to the ideal on $\omega_2$. Split it and code by the same method a subset of $\omega_2$. Assuming that those dense ideals exist for every regular $\kappa$ (which is consistent relative to a huge cardinal, by a theorem of Foreman), it is possible to continue with this method up to the first inaccessible. Foreman's theorem only gives us a $\kappa$-centered ideal on $\kappa$, but similar argument works for those ideals as well.

Split $C\setminus B$ into $\omega_1$ sets from $A$, up to an error in $I$, $\{F_i \mid i < \omega_1\}$. We assume that the collection $\{F_i \mid i < \omega_1\}$ is the first such partition in some fixed well ordering. We can assume that for every $i < j < \omega_1$, $F_i \cap F_j = \emptyset$ (using the $\sigma$-completeness). Code an element of $y \in \mathcal P(\omega_1)$ by coloring the components: $q\leq p$, $q_\alpha (n) = 1$ iff $\alpha \in F_i$ where $i \in y$. Now, every $y \in \mathcal{P}(\omega_1)$ is coded by a real from $V$ which in turn coded by a natural number.

The decoding is as follows: Let $G$ be the generic filter. Let $g_\alpha \colon \omega \to \omega$ be the generic real defined by $G$ for $\alpha < \omega_1$. For every natural $m$ we check if $n$ codes a real number in the means of Joel's answer. Assume that it does, and let $r_m$ be this real. By fixing a surjection from $2^\omega$ onto $A \times \omega$, $r$ corresponds to a couple $(C_m, n_m)$. Let $\{ F^m_i \mid i < \omega_1\}$ be the partition as above. Let: $$y_m = \{i < \omega_1 \mid \{\alpha \in F_i \mid g_\alpha (n) = 0\} = F_i \mod I\}$$. By density arguments, we know that for every $y\in \mathcal{P}(\omega_1)^V$ there is $m$ such that $y = y_m$.

Generalization: In order to generalize to higher cardinals (up to the first inaccessible), we will need to use $\kappa$-centered ideals on $\kappa$ (instead of $\kappa$-dense). An ideal $I$ is $\kappa$-dense if there are collection of $\kappa$ filters $\mathcal{F}_\alpha$, $\alpha < \kappa$ such that $I^+ = \bigcup_\alpha \mathcal{F}_\alpha$.

By a theorem of Foreman (appears in the Handbook, page 1047, Theorem 7.57), it is consistent relative to a huge cardinal that every regular cardinal $\kappa$ carries a $\kappa$-complete $\kappa$-centered ideal. Moreover, for accessible $\kappa$, those ideals (and also their restriction) are not $\kappa$-saturated, so every positive set $A$ can be partitioned into $\kappa$ positive sets.

Let's assume that in $V$ for every regular $\kappa$, there is $\kappa$-centered $\kappa$-complete non-trivial ideal. Assume also $GCH$, for simplicity.

We will show by induction on cardinals $\lambda$ less than the first inaccessible (if there is one), that the product of $\lambda$ copies of the Cohen forcing collapses $2^{\lambda}$. In fact we will construct the coding method (relatively) explicitly.

Case 1: $\lambda = \mu^+$. This is similar to the $\omega_1$ case above. Let $I$ be a $\lambda$-centered, $\lambda$-complete ideal on $\lambda$ as witnessed by $\{\mathcal{F}_i \mid i < \lambda\}$. We will generically code every element $y$ of $2^\lambda$ by an ordinal from $\lambda$ and a natural number (which in turn are generically coded by a natural number, by the induction hypothesis). Let $p$ be a condition and assume that $n$ is a natural number such that $C = \{\alpha < \lambda \mid n\notin \text{dom }p_\alpha\}\in I^+$. Assume that this set is in $\mathcal{F}_\gamma$. Since the difference between all two sets in $\mathcal{F}_\gamma$ is in $I$, we can pick an arbitrary element there and split it, as before, to $\lambda$ sets - $\{F_i \mid i < \lambda\}$. Again, those sets can be assumed to be disjoint and their union is $C$, up to an element from $I$. We continue, and extend $p$ to code $y$ in the generic in the same way as above: $q\leq p$, $q_\alpha(n) = 0$ iff $\alpha \in F_i \cap C$ and $i\in y$ so $y$ will be coded by $(\gamma, n)$ in the generic filter.

Case 2: $\lambda$ is singular. Let $\{\mu_i\mid i < \text{cf }\lambda\}$ be a cofinal sequence, $\mu_0 > \text{cf }\lambda$. We split the product into a product of $\text{cf } \lambda$ parts. It is clear that every element of $2^\lambda = \prod 2^{\mu_i}$ is coded by an element in $(\omega^{\text{cf }\lambda})^V$, but the latter is coded with a natural number, by the induction hypothesis.