This is a bit of an odd idea: you wish to keep the plant as it is but you'll like the roots to be closer to the fence, but it's ok.
Strange idea: dig up the plant with a large root ball no smaller than a football sized lump of soil and plant next to the fence as such because the bigger the rootball the better the plant will be. However, you mustn't damage the stems or over bend them as if they brake- then game over.
It does seem to be an odd place to plant a climbing plant so far away from a fence, isn't it? The only idea I can come up with is to just water a couple of days before you do it and allow to dry out a bit before you attempt it- a plastic sheet placed next to it as you move it will lessen the mess and hold the ball together better. However you might have to go quite deep to get the main root out whole- think maybe 2-3 feet deep- quite an undertaking but possible. Burying the stems will only rot the plant as they are not roots (those stems will not layer as they are too big and old).
If you think you're not quite up to the task, perhaps camouflaging the stems with some other plants might be an idea. Use something that would be evergreen and about a foot off the ground so one could hide those stems in plain sight or perhaps use the stems for something else to climb up and over into the fence too- like an annual flowering thing (sweet peas?). However you might have to put up a support for both- an arch or trellis with a hole in the middle to allow a view down the garden.