Gravitational Multipoles in General Stationary Spacetimes

I thank the referee for his remarks. I would be happy to adjust the wording of the abstract and the opening of section 3.1.2 to accommodate the remarks 2 and 3.

With respect to remark 1, it would certainly be nice to find a similar and less technically obscure gauge-fixing condition on the improvement form compared to the one given in the paper. However, unfortunately, I don't think the referee's suggestion is applicable.
The manifold in question does not have a boundary (or, at most, it has only a boundary at small r, but is unbounded for large r), so the formalism of 1904.12869, and in particular, the unique (Helmholtz) decomposition (2) in the referee report, does not hold. For example, in a standard counterexample to the uniqueness of the Helmholtz decomposition in non-compact spaces without boundaries, one can shift psi -> psi' = psi + H and phi->phi'=phi-H, where H is harmonic (d*dH = 0), and it can be seen that the decomposition (2) also holds with \delta psi' = 0 and orthogonality of psi' and phi'. (Such harmonic functions also clearly exist and can be non-trivial, eg. 1/r or its derivatives.)
Similarly, the demand that the improvement form would be orthogonal to (all) exact forms is not enough to fix the gauge conditions as given in the paper draft, as shifting such an improvement form by dH (with H harmonic) would still be allowed.
Incidentally, it is tempting to attempt a similar reasoning using a Helmholtz/Hodge decomposition of the improvement form but instead on the compactified manifold \tilde M, since on this compact manifold such Hodge decompositions are unique. However, to be able to do this would first require knowledge of how the improvement one-form transforms under conformal transformations. I strongly suspect that (again, unfortunately) the improvement one form does not transform in a simple way under such conformal transformations. (At least, there is no particular reason for it to transform simply.)