I can't quite say about the BCT question, it's not immediate to me. However, for the Rasiowa–Sikorski lemma the answer is negative.

Recall that Dependent Choice is equivalent to the statement "Every tree of height $\omega$ without maximal nodes has an infinite branch". Note that if $T$ is a tree, then a generic filter is a branch, i.e. a maximal chain. But now, if $T$ is a tree of height $\omega$ and the R–S lemma holds, then $D_n=T\setminus T\restriction n$ (that is, all the nodes of height $>n$) is a countable family of dense open sets, and a generic is a branch. Therefore Dependent Choice must hold.

This, along with the result you mention (which I can't verify at the moment) suggest that BCT for compact Hausdorff spaces is equivalent to $\sf DC$ over $\sf ZF+BPI$. Of course, the interesting question is, since we know that $\sf DC$ and $\sf BPI$ are entire independent of each other, is there a model where BCT for compact Hausdorff spaces hold, but $\sf DC$ fails? Or rather, what is that model?

The key observations are that BPI is equivalent to the Stone representation theorem for Boolean algebras, and that for the Rasiowa–Sikorski lemma we can focus on [complete] Boolean algebras, since they are forcing equivalent (so we can restrict the generality of partial orders).

Now, one implication is a consequence of ZF. Since RS is equivalent to Dependent Choice, which itself is equivalent to the downward Löwenheim–Skolem theorem, one can just use that argument.

Alternatively, if $X$ is a compact Hausdorff space and $D_n$ are dense open sets, and without loss of generality $D_{n+1}\subseteq D_n$. take $U$ to be a non-empty open set, and consider the forcing whose conditions are sequences $(x_i,W_i)_{i<n}$ such that $x_i\in D_i\cap U$ and $W_i$ is open such that:

1. $x_i\in W_i\subseteq \overline W_i\subseteq U\cap D_i$, and
2. $\overline W_i\subseteq W_j$ if $j<i$.

Now consider $E_n$ to be the dense open set in the forcing whose conditions are sequences of length at least $n$. By the Rasiowa–Sikorski lemma there is a generic meeting all of these $E_n$s which defines a sequence $(x_i,W_i)_{i<\omega}$. Now observe that $\{\overline W_i\mid i<\omega\}$ is a family of compact sets with a finite intersection property, therefore their intersection is non-empty, and it contains a point in $\bigcap D_i\cap U$ as wanted.

In the other direction we need to use BPI, and we use it in the form of Stone's representation theorem. Given a notion of forcing, we may assume without loss of generality that it is a complete Boolean algebra $B$ and we can consider its Stone space, $S(B)$, the space of all the ultrafilters on $B$.

If $D\subseteq B$ is a dense open set, then $D^*=\{F\in S(B)\mid\exists b\in D, b\in F\}$ is a dense open set in $S(B)$. Therefore, by the BCT for compact Hausdorff spaces, if $D_n$ is a sequence of dense open subsets of $B$, $\bigcap D^*_n$ is dense, and in particular not empty. Take any $G\in\bigcap D_n^*$, then for all $n<\omega$, $G\cap D_n$ is non-empty as wanted.