How to pick what coordinates transformations to apply in general relativity?

First, allow the cosmological constant to be expressed via the Hubble constant as $$frac{Lambda}{3c^2}=H_0^2 tag{1}$$ Then the metric is called the static form of the de Sitter metric. The coordinate transform that turns a static metric to a metric in comoving frame is $$dtau=dt-frac{v}{c^2} left(1-frac{v^2}{c^2}right)^{-1}drtag{2}$$ where $tau$ is time in comoving frame. The velocity of comoving frame in this particular case is $$v=H_0 rtag{3}$$ The substitution of (2) with the use of (1)&(3) into the given metric leads to the Gullstrand–Painlevé (GP) form $$ds^2=-c^2left(1-frac{v^2}{c^2} right)dtau^2- 2 v dtau dr+dr^2+r^2 dΩ^2 tag{4}$$ Basically this is the metric in comoving frame. $v$ is given by (3) and $tau$ is time in comoving frame. However, $r$ is still coordinate distance. Therefore, in order to to use comoving distance one may resolve (3) for $r(tau)$ which yields $$r=R : e^{H_0tau} tag{5} quad mathrm{and} quad v=R :H_0 :e^{H_0tau}$$ where the arbitraty constant of integration was set to $R$ which is the comoving distance. Using $$dr= dR : e^{H_0tau} +v : dtau tag{6}.$$

The substitution of (6) into (4) results in $$ds^2=-c^2 dtau^2+ left(e^{H_0tau}right)^2left(dR^2+R^2 dΩ^2right)tag{7}$$ which is the FLRW form for the spatially flat metric with the cosmological scale factor $a(tau)=e^{H_0 tau}$.

You may check Tolman for the de Sitter case or this paper https://www.nature.com/articles/srep00923 for general correspondence of static and comoving metrics.