= Solution
Choose reference depth $h_0$, <velocity> $u_0=\sqrt{gSh_0/f}$ and <concentration> $c_0=E(u_0)$. Set the adaptation length $L=h_0u_0/v_s$ and the slow bed-evolution time $t_b=\rho_s(1-\phi)h_0/(v_sc_0)$. Scale both bed and surface elevation with $h_0$. Then
$$
\boxed{\epsilon=\frac{L}{u_0t_b}=\frac{c_0}{\rho_s(1-\phi)},\qquad F^2=\frac{u_0^2}{gh_0},\qquad\delta=\frac{SL}{h_0},\qquad E^*(u)=\frac{E(u_0u)}{c_0}.}
$$
These choices balance <sediment> advection with settling, bed mass change with exchange, and reference drag with downslope gravity. Direct substitution gives
$$
h=\eta-s,\quad\epsilon h_t+(hu)_x=0,\quad
F^2(\epsilon u_t+uu_x)=-\eta_x+\delta(1-u^2/h),
$$
$$
\boxed{h(\epsilon c_t+uc_x)=E^*(u)-c=-s_t.}
$$
Here $E^*(1)=1$ by construction. The small parameter $\epsilon$ is the suspended solid mass relative to bed solid mass at the chosen scales, explaining the separation between fast fluid motion and slow morphology.
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