Regarding what you wrote *I would be equally happy with a proof of Hopf's theorem instead*, it seems that the very fact that there is "conservative part" for a nonsingular action is unavailable for general lcsc groups. However, there is a "version" of the Hopf Decomposition for the purpose of checking the divergence
$$(*)\quad \intop_{G}\frac{d\mu\circ g^{-1}}{d\mu}f\left(gx\right)dm\left(g\right)=\infty.$$

This can be done as follows. Pick a lattice in $G$ (if there exists any!) and take the Hopf Decomposition $D\cup C$ w.r.t. this lattice (regarding this lattice as an acting group on its own right, and using the Hopf Decomposition for the countable case as in [1, Proposition 1.6.2]). It is then true that $(*)$ holds for $x\in D$ and, moreover, the set $D$ is independent on the lattice up to null set. See [1, Theorem 1.6.4] and [2, Corollary 2.2].

**References**

[1] <cite authors="Aaronson, Jon">_Aaronson, Jon_, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs. 50. Providence, RI: American Mathematical Society (AMS). xii, 284 p. (1997). [ZBL0882.28013](https://zbmath.org/?q=an:0882.28013).</cite>

[2] <cite authors="Roy, Parthanil">_Roy, Parthanil_, [**Nonsingular group actions and stationary S\(\alpha \)S random fields**](http://dx.doi.org/10.1090/S0002-9939-10-10250-0), Proc. Am. Math. Soc. 138, No. 6, 2195-2202 (2010). [ZBL1196.60093](https://zbmath.org/?q=an:1196.60093).</cite>