On statistical bases in Banach spaces
This is an open problem due to Vladimir Kadets. I would not expect an easy answer here. The problem with statistical convergence is that this filter is not countably generated (in which case these functionals would be continuous).
A more general question is also open. Suppose that $\mathcal{F}$ is a filter on $\mathbb N$. Say that a sequence $(e_n)$ in a Banach space $X$ is an $\mathcal F$-basis if every element $x\in X$ has a unique expansion as $$\mathcal{F}-\lim_{n\to\infty} \sum_{k=1}^n e_k^*(x)e_k.$$ Are the linear functionals $(e_n^*)$ continuous?
The only positive result so far is a theorem of Kochanek which says that the answer is yes when $\mathcal F$ is generated by less than $\mathfrak p$ elements (here $\mathfrak p$ denotes the pseudo-intersection number; see this entry on T. Gowers' blog for more details).
T. Kochanek, ℱ-bases with brackets and with individual brackets in Banach spaces, Studia Math. 211 (2012), 259-268.
For the statistical convergence, one may suggest a meta-proof. The definition of statistical convergence does not involve (full) Axiom of Choice. There are models of ZF+DC where all linear functionals on Banach spaces are continuous hence you cannot have a constructible counterexample, should it exist.
In a recent note with J. Swaczyna (arXiv:2005.04873), we proved that assuming the existence of certain large cardinals (for example, the existence of a super-compact cardinal) that filter bases whose underlying filter is a projective subset of the Cantor set, have continuous coordinate functionals. The role of large cardinals is to make the heuristic proof I outlined in my other answer work.
The filter of statistical convergence is actually $F_{\sigma \delta}$ (hence Borel, hence projective), so in a theory stronger than ZFC, the answer to your question is affirmative. We still expect that, at least for Borel filters, the question about continuity of coordinate functionals should have a positive answer in ZFC. However, our proof method does not give a chance to invoke Schoenfield's absoluteness theorem in that context.