If I have an exterior of OSB3 with breathable membrane over and a decent roof overhang of 500mm, do I still need to baton on top of that before installing treated shiplap that I would stain both sides? Is it vital for the membrane to drip moisture, or to have that air gap?
You should not use breathable membrane for exterior walls- It is to allow a roofspace to breathe. An air permeable membrane can allow moisture into your structure from the outside air. Given that batten is not expensive, and allows air circulation behind the cladding preventing damp/mould/rot, why would you not use it?
?? Yes, use battens to create a ventilation space. Yes use a breather membrane under the cladding e.g. https://www.dupont.co.uk/solution-finder/results.html?BU=pbs&tagId=DuPont:type/breather-membrane or https://www.roofingsuperstore.co.uk...anes/timber-frame-wall-breather-underlay.html Also, have a look at a cement board plank cladding - e.g. hardieplank or cembrit. Bit more expensive material, but no staining required, and maintenance free.
Sorry, usually breather membrane means the roof stuff. Tyvek housewrap is as Mr Rusty indicates, the right stuff