Applying $\epsilon=b_i(t)x^i$ to $A_\mu=0$ gives
$$
A_0=\dot b_i(t)x^i,
\qquad
A_i=b_i(t).
$$
Its field strength vanishes, so it is a <large gauge transformation> of the background. To arise as the zero-momentum limit of a physical transverse perturbation, $b_i(t)$ must obey the zero-momentum vector equation of motion. In the variables used in the question this is
$$
\boxed{\partial_t(a^3\dot b_i)=0},
$$
with a constant growing solution and a decaying solution proportional to $\int^t dt'/a^3(t')$. This condition makes the large gauge mode an <adiabatic mode> that can be continued to small nonzero momentum.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 312 / 2 / c / Solution

A large gauge transformation does not approach the identity at the relevant spatial or asymptotic boundary. Although it leaves the local field strength unchanged, its boundary charge can act nontrivially on physical states and constrain soft limits through a <Ward identity>.

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