There's no adverbial trickery going on here. Indeed in a normal sentence it would be something like 木の家で暮らそう, or "let's live in a wood house." Taken at face value it may seem like it's just saying that you should live with a tree or 'with' a wood house (like a tree as a roommate, as you say), but this is advertisement speak. The effect of saying it this way is ultimately emphasis on the woodiness of the house. It is telling us to live with the wood house in a deeper sense of the word. My opinion would be that saying と evokes more of an image of living 'naturally' with your house rather than living in it, that is, with natural materials rather than something synthetic. I don't know how many would share that opinion, so take it for what it is! Chocolate's comment above seems to corroborate this, too. It's a way of placing value on wood, on trees, and invoking a consciousness of that in your daily life.
But at any rate it's not using 木の家 as an adverb. It just wouldn't really make sense that way, or at least it makes less sense than the alternative interpretation. If all else fails defer to Occam's Razor here.
The translation question is probably off topic considering there isn't going to be a right answer for it. The idiosyncratic nature of the phrase makes a translation that captures the nuances in the same way impossible and would probably be left up to a marketing team rather than translators to choose an alternative (I guess, anyway!). Because there is no solid equivalent every situation will require a different translation that captures that sentence's unique meaning.