First off, the cart path is NOT a local rule - I think this has been discussed before - see obstructions at the bottom. As for the ruling, you can not take relief inside a hazzard:

24-2. Immovable Obstruction
b. Relief
Except when the ball is in a water hazard or a lateral water hazard, a player may take relief from interference by an immovable obstruction as follows:
(i) Through the Green: If the ball lies through the green, the player must lift the ball and drop it without penalty within one club-length of and not nearer the hole than the nearest point of relief. The nearest point of relief must not be in a hazard or on a putting green. When the ball is dropped within one club-length of the nearest point of relief, the ball must first strike a part of the course at a spot that avoids interference by the immovable obstruction and is not in a hazard and not on a putting green.
(ii) In a Bunker: If the ball is in a bunker, the player must lift the ball and drop it either:

(a) Without penalty, in accordance with Clause (i) above, except that the nearest point of relief must be in the bunker and the ball must be dropped in the bunker; or
(b) Under penalty of one stroke, outside the bunker keeping the point where the ball lay directly between the hole and the spot on which the ball is dropped, with no limit to how far behind the bunker the ball may be dropped.
(iii) On the Putting Green: If the ball lies on the putting green, the player must lift the ball and place it without penalty at the nearest point of relief that is not in a hazard. The nearest point of relief may be off the putting green.
(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.

The ball may be cleaned when lifted under this Rule.
(Ball rolling to a position where there is interference by the condition from which relief was taken — see Rule 20-2c(v).)
Exception: A player may not take relief under this Rule if (a) it is clearly unreasonable for him to make a stroke because of interference by anything other than an immovable obstruction or (b) interference by an immovable obstruction would occur only through use of an unnecessarily abnormal stance, swing or direction of play.
Note 1: If a ball is in a water hazard (including a lateral water hazard), the player may not take relief from interference by an immovable obstruction. The player must play the ball as it lies or proceed under Rule 26-1.
Note 2: If a ball to be dropped or placed under this Rule is not immediately recoverable, another ball may be substituted.
Note 3: The Committee may make a Local Rule stating that the player must determine the nearest point of relief without crossing over, through or under the obstruction.

Obstructions
An “obstruction’’ is anything artificial, including the artificial surfaces and sides of roads and paths and manufactured
ice, except:
a. Objects defining out of bounds, such as walls, fences, stakes and railings;
b. Any part of an immovable artificial object that is out of bounds; and
c. Any construction declared by the Committee to be an integral part of the course.
An obstruction is a movable obstruction if it may be moved without unreasonable effort, without unduly delaying play and without causing damage. Otherwise it is an immovable obstruction.

__________________
Gord

Quote of the month:
"It's easy to see golf not as a game at all but as some whey-faced, nineteenth-century Presbyterian minister's fever dream of exorcism achieved through ritual and self-mortification." ~Bruce McCall