Very unlikely you would get an overlength permit. Overlength permits are when hauling an item that cannot be legally towed at a legal length. An example a lot of people have seen are the parts for the large air turbines. They're made in one piece and cannot be shortened to legal length. Therefore they would qualify for an overlength permit. A double tow would not qualify for an overlength permit because double towing is not a requirement. Either the 5er or the following trailer can be disconnected and towed separately at legal length.
What a lot of people do not also understand about permit movements is for oversize loads (height, weight, length) those hauling it do not get to drive wherever and whenever they want. They are a 1 time permit to move the item from 1 specific location to another specific location. The permit will have hours of movement. The permit will have the exact routes and roads the vehicle can travel. No exceptions for any of that. Can't slide into a Walmart just because you want a potty break. That would be an off route movement, permit no longer valid, and heap big fines in addition to the permit being revoked.
Additionally, depending on the size of the load it may require an escort vehicle, properly licensed and equipped. If it's a really large load it may also require a police escort.
None of that is going to be cheap. But none of that is relevant anyway because you could not get a permit for double towing.