fnufan
I am going to play the devil's advocate here. I am not a rules official, but I know several rules officials and one piece of advice that I received from one of them was that the answer to most rules questions can be found in the Definitions section of the Rules of Golf. So I will offer up this definition to start with:
Teeing Ground
The "teeing ground" is the starting place for the hole to be played. It is a rectangular area two club-lengths in depth, the front and the sides of which are defined by the outside limits of two tee-markers. A ball is outside the teeing ground when all of it lies outside the teeing ground.
So it would seem that if a cart path is within two club-lengths of the two tee-markers, then it is still within the "teeing ground" as defined by the rules. Of course, a cart path is also an immovable obstruction where relief is possible - including relief under Rule 24-2b. iv):
On the Teeing Ground: If the ball lies on the teeing ground, the player must lift the ball and drop it, without penalty, in accordance with Clause (i) above.
So "Clause (i) above" allows relief within one club-length of the NPR. Whether that is in the fairway/rough of the 9th/18th hole is impossible for us to determine online, but i think some sort of relief would be possible. At the very least, I think the OP could choose to tee the ball on the flat, even cart path (if he could get a tee into the ground) and play from there.