The law eludes that a governmental entity cannot post a 30.06 sign and it be valid(which isn't the case so that doesn't apply), but a school is off limits no matter what. So I'd have to say that this situation is one where the heat stays in the car.
Look at it this way. If there were an adjoining door to a courthouse that lead into a Dunkin' Doughnuts because the judge likes his breakfast to be close at hand, that place would be off limits. The business doesn't have to post since there is an adjoining entrance through to the courthouse.
Since the facility is connected with an adjoining door, a reasonable person would consider that to be part of the entire building. Remember, we are dealing with "reasonable" people. Not always the same as "sane".
