Per the AIM, you should be able to receive the local AWOS/ASOS observation within 25 miles of the airport below 10,000 feet. At that time, you should set your altimeter to that local setting as opposed to the one issued by the controller. Most approach controls will have their radar display set up to advise them of the altimeter setting of the primary airport.
Mode C transponders broadcast your pressure altitude, or the altitude you'd be at if your altimeter were set to 29.92." The radar system takes that pressure altitude, applies the local altimeter setting to it, and provides the controller with an accurate depiction of the altitude you see on your altimeter. If you were to switch to a local altimeter that was vastly different than that of the primary airport, your altitude will change on the controller's radar display, but simply informing the controller that you have received the "one-minute" weather at your destination should alleviate his or her confusion, if there is any at all.