NFA-$\epsilon$ extended transition function for inverted strings
First, there's a typo at your definition of $\hat\delta$. Case $\hat\delta(q,\alpha x)$ should be $\bigcup_{p_i \in \hat\delta(q,\alpha)}ECLOSURE(\delta(\color{green}{p_i},x))$ (green for highlighting the difference).

The equivalent definition in terms of a string given as $(x\alpha)$, where $x\in\Sigma$ and $\alpha\in\Sigma^*$, is
$$
\hat\delta(q,x\alpha) = \bigcup_{p_i \in ECLOSURE(\delta(q,x))} \hat\delta(p_i,\alpha)
$$