You could use a battery pack. Thunderbolt simply does not provide as much bus power as FireWire did. Part of the power is consumed by the FireWire controller and the thunderbolt controller in the adapter; there is only about 7-7.5W of power available on the output of the Thunderbolt adapter; the ULN-2 requires at least 8 Watts; so there is no way to power the ULN-2 from the TB->FW adapter. So -- just to be clear -- it is not the ULN-2 -- it is the TB adapter; as you saw, things worked as expected on the native FW port.
Depending on how long you want to run for via battery, you can work with a pretty small battery pack; for example, if you get a 9v Lithium battery (which is the size of a 9v), you can probably power the ULN-2 for about 1-1.5 hours.