Could you lower the whole setup (including the bees clinging to the bottom) into a larger cardboard box? Those bees on the bottom will probably move around as you gently lower them into the cardboard, but if they fly you could wait for them to settle back in. Then, close the cardboard flaps, tape the flaps, and transport them to their permanent site...once there, all you'd have to do is lift the hive box from the cardboard carrier and, viola!, they're home.
Liz