Cardinality of a Rosenthal compact space?

Let $X$ be a polish space a real valued function $f$ on $X$ is of the first Baire class if $f$ is a pointwise limit of a sequence of continuous functions on $X$. Let $B_{1}(X)$ denote the space of all real-valued first Baire class functions on $X$, endowed with the topology of pointwise convergence. A compact space $K$ is called Rosenthal compact if $K$ can be embedded in the space $B_{1}(X)$ for some polish space $X$.

I am Trying to show that every Rosenthal compact space has the cardinality and weight at most $2^{\omega}$. Thank for any help about any of this two questions.

I did not succeed to construct an injective map from $K$ to any set of cardinality $2^{\omega}$ for instance.

$X$ is separable, so $|C(X)|\le 2^\omega$. Thus, there are at most $(2^\omega)^\omega=2^\omega$ sequences of continuous real-valued functions on $X$, and therefore $|B_1(X)|\le 2^\omega$. $B_1(X)$ is a subspace of the product of $|X|$ many copies of $\Bbb R$, and $|X|\le 2^\omega$, so $B_1(X)$ is a subspace of $\Bbb R^{2^\omega}$, which has weight $2^\omega$. It follows that any space embedded in $B_1(X)$ has cardinality and weight at most $2^\omega$.