I also initially had the tongue-and-groove option in mind, but then decided against it because of the price.
Intuitively, I would have now stapled the underlay membrane to the battens. This way, the rhombus battens lie tight against the underlay membrane and the wall behind can breathe. Possibly, I would have wrapped the lathing of the substructure with an additional layer of underlay membrane to protect it from moisture.
How is it done correctly then?
From what you usually find online, those without insulation have not installed an underlay membrane either.