$\mathsf{RegExpEq_*} \in \mathsf{coNPC}$ but why isn't in $\mathsf{P}$
By toggling the accepting states of $M$ we can construct an NFA which accepts $\overline{L(R)}$.

Well, that's not true. You won't get $\overline{L(R)}$. That would have happened if $M$ was a DFA, rather than a NFA. So you have to first convert $M$ to $M'$ which is a DFA, but then you would have to spend an exponential time there.