As much as I love natural things, 30 years down the road of projects in a rural location I've done
enough redoing on sheds, a dock, wooden window frames, decks, grapevine poles. It's inevitable. I don't hesitate anymore to use treated wood in an exterior application (the dock, a shed in contact with soil, wooden
fence posts) because if the wood is bare it may only last a few years. I'll probably get more brain tumors from sitting in front of a computer screen :) The cost of replacing wood is expensive. It's probably tripled since I first did the projects. I don't mean to say that treated wood should be used in a building frame, it shouldn't, just when in contact with soil.
A solid-color polyurethane stain will penetrate, and while it isn't perfect, it gives some protection without too much unnatural stuff. There will also be plenty of cob around it, with
straw that absorbs any off-gassing. I would be more concerned about the off-gassing of synthetic carpets. There's also a mold additive at the paint store that can be added to paint and stain, then applied. The shady side of a building (north in the northern hemisphere) will be more vulnerable to moisture/mold/moss. If you have a shed or a house there now, check the roof and see what's growing on it if it's been there 5+ years.
At some point there's going to be a weak spot where blowing rain gets to the cob, or wet and eventually dripping snow, which means dampness gets to the wood. Or dampness coming up from the soil will get into the wood, into the interior. There will be a lot of tweaking, unless the building is in the desert.
Not everyone has termites, but if they are around, they are quick! If wood stays damp enough it will get moldy.