Have they given you an idea of how long it will take?
I think you'd be putting them in the shade anyway? thats about all I can suggest, could you actually plant them in the shade and replant near the house in Autumn or spring rather than leaving them wrapped up or in pots?
Just a thought, are you lifting your clematis and Virginia Creeper? I know they would be heavy but would it be possible to cut them to say, 4ft, leave them entwined, lifting both clumps together? sorry if that sounds silly, don't know your situation or how old they are and just wondering if it would save a bit of growth? I'd be worried it wouldn't recover at this time of year?
I joke that when we cleared my late fathers garden, about 4 years ago, the shrubs grabbed hold of my fork and wouldn't let go, there was about 6" soil at the most, then solid clay, waterlogged as well i think. the roots hadn't got into the clay. Brought them up to where I live, the holly I had planted there over twenty years ago, looked nearly dead, lifted and replanted here is now full of life, the others have all done well as well, that was after the planting season, just had to keep them well watered that year.