If I understand correctly the einbein can be understood as the component of a one-dimensional metric, along a particle worldline. I'm not so sure of writing it out explicitly, but this seems like it might be: $$-c^2 d\tau^2 = - c^2 e^2 dt^2 \ .$$ To me this seems mathematically and conceptually close to the idea of a threading lapse function.
Edit (explanation of the lapse functions): A general metric can be written in the form \begin{equation} ds^2 = g_{xx}\, dx^2 + 2 g_{tx}\, dt\, dx + g_{tt}\, dt^2\ . \end{equation} There are two ways we can "complete the square": the space orthogonalization which corresponds to the formalism variously called ADM, 3+1 or "slicing" is \begin{eqnarray} ds^2 &=& g_{xx}\, (dx + \frac{g_{tx}}{g_{xx}}\, dt)^2 + (g_{tt}-\frac{g_{tx}^2}{g_{xx}})\, dt^2 \nonumber\\ &\equiv& g_{xx}\, (dx + N^x\, dt)^2 -N^2\, dt^2\ , \end{eqnarray} whereas the time orthogonalization corresponds to the 1+3 or threading formalism is \begin{eqnarray} ds^2 &=& (g_{xx} -\frac{g_{tx}^2}{g_{tt}})\, dx^2 + g_{tt}\,(dt +\frac{g_{tx}^2}{g_{tt}}\,dx)^2 \nonumber\\ &\equiv& \gamma_{xx}\, dx^2 -M^2\,(dt +M_x\,dx)^2\ \end{eqnarray} where $M$ is the threading lapse function whereas $N$ is the slicing lapse function.
We may also understand the threading lapse function relates the flow of an arbitrary parameter $t$ of a congruence line to the flow of the proper time $\tau$ of the fluid element moving along that line, $\frac{d\tau}{dt} =: M$. The slicing lapse function $N$ can be understood to relate the flow of an arbitrary parameter $t$ of an integral curve of tangent vector $n$ to the flow of the proper time of the (Eulerian) observers moving along that line. See e.g. the appendices of this paper.
