Codings of separable compact subsets of the first Baire class

Let $X$ be a Polish space and $K$ a separable compact subset of the first Baire class on $X$. For every sequence $\text{bs}$ dense in $\text{kk}$, the descriptive set-theoretic properties of the set $$\text{lbf}=\{L\in [\text{nn}]:(f_n{)}_{n\in L}\text{is pointwise convergent}\}$$ are analyzed. It is shown that if $K$ is not first countable, then $\text{lbf}$ is ${\text{PB}}_1^1$-complete. This can also happen even if $K$ is a pre-metric compactum of degree at most two, in the sense of S. Todorcevic. However, if $K$ is of degree exactly two, then $\text{lbf}$ is always Borel. A deep result of G. Debs implies that $\text{lbf}$ contains a Borel cofinal set and this gives a tree-representation of $\text{kk}$. We show that classical ordinal assignments of Baire-1 functions are actually ${\text{PB}}_1^1$-ranks on $\text{kk}$. We also provide an example of a ${\text{SB}}_1^1$ Ramsey-null subset $A$ of $[\text{nn}]$ for which there does not exist a Borel set $B\supseteq A$ such that the difference $B\setminus A$ is Ramsey-null.