= Solution
Let
$$
H_n=\mathbb E_A T_{BC}
$$
for the walk on $G_n$. Decompose $G_n$ into its three outer copies of $G_{n-1}$ and trace the walk only when it passes between distinct corner vertices of these copies. The <self-similarity> and reflection symmetry of the Sierpinski graph make this trace the simple random walk on $G_1$. It requires an expected five transitions to hit the two target outer corners.
Each coarse transition is an excursion across a copy of $G_{n-1}$ and has mean duration $H_{n-1}$. Applying the strong Markov property at the coarse stopping times gives
$$
H_n=5H_{n-1}.
$$
Since $H_1=5$, induction yields
$$
\boxed{\mathbb E_A T_{BC}=H_n=5^n}.
$$
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