You can move the dirt with a skid steer without causing too much damage if the ground is dry. You have to consider how many trips you'd have to take through whatever access you do have as you will eventually cause ruts or soil compaction that will show for some time. I had to Bobcat a pool's worth of soil out after digging, because there was no room for a truck. It took two days. I gave that dirt away as fill (it was junk) and then spent another day moving another thirty yards of pulverized soil in so I could build beds around the pool.
Your alternatives are limited, though LOOSESTRIFE has the all around best option. Blowing the soil would be awesome. The only problem is who has one? I imagine it would be pretty expensive to hire, as the smallest one costs $120,000 to purchase.
Another option: Is there better access through the duplex? Perhaps you can obtain written permission and bring triax dump trucks in that way.