Computational complexity of pi
OK, James Lee has pointed me to this 2011 paper by Samir Datta and Rameshwar Pratap, which proves that my language $L$ (encoding the digits of $\pi$) is in the fourth level of the counting hierarchy ($\mathsf{PH}^{\mathsf{PP}^{\mathsf{PP}^{\mathsf{PP}}}}$; thanks to SamiD below for pointing out a missing $\mathsf{PP}$ in the paper, which I'd simply repeated in my answer!).  The paper also explicitly discusses my question of lower bounds on the complexity of computing the binary digits of irrational numbers, though it only manages to prove a very weak lower bound for computing the binary digits of rational numbers.  This is exactly what I was looking for.

Update (April 3): An amusing consequence of the digits of $\pi$ being computable in the counting hierarchy is as follows.  Suppose that $\pi$ is a normal number (whose binary expansion converges quickly to "effectively random"), and suppose that $\mathsf{P} = \mathsf{PP}$ (with the simulation involving only a small polynomial overhead).  Then it would be feasible to program your computer to find, for example, the first occurrence of the complete works of Shakespeare in the binary expansion of $\pi$.  If that sounds absurd to you, then maybe it should be taken as additional evidence that $\mathsf{P} \ne \mathsf{PP}$. :-)