= Solution
The <mass-shell condition> and $P_\mu^c=-mg_{\mu\nu}\dot x^\nu$ imply $g_{\mu\nu}\dot x^\mu\dot x^\nu=-1$, so the affine parameter is proper time up to orientation. Choose its future orientation. With <conformal time> $t$,
$$
-a^2\dot t^2(1-\mathbf v^2)=-1,\qquad
\dot t=\frac\gamma a,\qquad \dot x^i=\frac\gamma a v^i.
$$
In a local comoving orthonormal frame both time and spatial coordinate intervals acquire the same factor $a$, so the measured peculiar velocity is precisely $\mathbf v=d\mathbf x/dt$. Thus the <physical momentum> obeys
$$
\mathbf P=\gamma m\mathbf v=-\frac{\mathbf P^c}{a},\qquad
\boxed{|\mathbf P|\propto a^{-1}.}
$$
The fixed vector direction also gives $\mathbf P(t)=a(t_*)\mathbf P(t_*)/a(t)$. With the conventional opposite overall sign for the affine particle action, the familiar <comoving momentum> is $\mathbf q=a\mathbf P=\mathbf P^c$ instead. The redshift law is identical under either sign convention.
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