We had agricultural PP for ours, but it is commercial size and sits on Section 75 (=agricultural only) land. We had no problems. If you use metal foot plates rather than concreting in the supports the tunnel cannot be classified as permanent, and it can be moved if required (with loads of digging...)
Caroline - yours sounds actually not that big, I would just go for it, plant a willow hedge or similar fast growing if it is visible from the road. And you are not in a National Park or similar?
But it is hit-n-miss with planning authorities at the moment, here they are falling over themselves to grant ANYTHING at the moment, just to keep themselves in a job....