You could end up breaking laws in every State between Michigan and DC. The federal law designed to allow you to transport a firearm through States where your permit is invalid cannot be used. It only applies if you are allowed to possess and carry at your point of origin, your destination, and all stops in between. Since you are not allowed to possess and carry in DC, you could be violating the laws of any State you pass through along the way in which your permit is not reciprocated.
IANAL, but watch out!
[I just thought of this: VA has fairly relaxed gun laws. Maybe you could make your trip to northern VA (check your permit situation), making that your destination. Leave your handgun in VA when you make your trip(s) into DC.]