With an Virginia non-resident permit you will be in violation of the federal GFSZ Act if you get within 1,000 feet of a school. The federal exemption only applies to those with a permit from the state in which the school zone is located:
18 USC 922(q)2(A)
It shall be unlawful for any individual knowingly to possess a firearm that has moved in or that otherwise affects interstate or foreign commerce at a place that the individual knows, or has reasonable cause to believe, is a school zone.
(B)Subparagraph (A) does not apply to the possession of a firearm—
(i)on private property not part of school grounds;
(ii)if the individual possessing the firearm is licensed to do so by the State in which the school zone is located or a political subdivision of the State, and the law of the State or political subdivision requires that, before an individual obtains such a license, the law enforcement authorities of the State or political subdivision verify that the individual is qualified under law to receive the license;
This may or may not be of concern to you but it is something of which you should be aware.