= Solution
Holding the background metric fixed, the linearized homogeneous-field equation is
$$
\delta\ddot\phi+3H\delta\dot\phi-\frac1{a^2}\nabla_{\rm com}^2\delta\phi+V''(\bar\phi)\delta\phi=0.
$$
Consequently each comoving <Fourier coefficient> obeys
$$
\boxed{\delta\ddot\phi_{\mathbf k}+3H\delta\dot\phi_{\mathbf k}+\left(\frac{k^2}{a^2}+m_{\rm eff}^2\right)\delta\phi_{\mathbf k}=0,\qquad m_{\rm eff}^2=V''(\bar\phi)}.
$$
The unscaled Laplacian in the starting equation must be interpreted as the physical spatial Laplacian; for the explicitly comoving <wavevector> it gives $k^2/a^2$, not $k^2$. Metric fluctuations have been discarded as permitted. The mode written later is a massless, minimally coupled approximation, requiring $|m_{\rm eff}^2|\ll H^2$; it is not a solution for an arbitrary potential <curvature>.
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