I moved a 37 high cichlid tank 150 miles in a uhaul truck in February in Iowa. I left the substrate in with enough water so that only the top of the gravel was exposed, put that in the back of the truck, and put the fish in a 5g bucket and set it on the floor of the passenger seat. I didn't use a heater or airstone, just made sure the floor heat was on enough to maintain temp and they all survived just fine, and I didn't have to cycle.
The other thing you could do is scoop out all the substrate into a bucket w/water over the top right before you move it, and get it back into the tank right away after the move. This would probably be the best considering the stairs.
If you can, get a few more 5g buckets and move the water too to minimize stress on the fish. Do it all quickly to minimize bacteria loss from low O2