= Solution
The spatial momentum operator acts only on the plane wave:
$$
\frac\hbar i\nabla\left[a^{-1}A_{\mathbf k}e^{i\mathbf k\cdot\mathbf x-ikt}\right]
=\hbar\mathbf k\left[a^{-1}A_{\mathbf k}e^{i\mathbf k\cdot\mathbf x-ikt}\right].
$$
Thus \b[the modes have fixed canonical/comoving wavevector momentum, not fixed physical momentum]. In the usual positive-energy field-mode convention the local <physical momentum> and energy are $\hbar\mathbf k/a$ and $\hbar k/a$. This agrees with conserved spatial <comoving momentum> and cosmological redshift proportional to $a^{-1}$.
The overall particle-action sign noted in (i) reverses the literal worldline canonical label relative to this usual mechanical momentum: it is $\mathbf P^c=-a\mathbf P$ for a future-directed particle with that printed action. The conserved plane-wave label therefore has the opposite sign if it is identified with that literal worldline label. This sign convention does not change the wavenumber magnitude or either conservation/redshift conclusion.
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