technically, the rulebook only says: "Monsters treat invisible characters exactly as if they were obstacles."More than that, being treated as obstacles is not actually a property of invisibility. Enemy figures are always treated as obstacles for purposes of movement except that flying creatures cannot land on them. Invisibility just makes figures unfocusable and untargetable. The interaction of these facts is where we get the "monsters treat invisible as obstacles" thing from.
It does not say that Characters can treat invisible monsters as if they were obstacles.
Perhaps a better wording for the rulebook would have been "Monsters treat invisible characters exactly as if they were obstacles in terms of movement and finding focus."
Knowing this, the answers to common questions like OP's pretty much trivially follow.