Do a test. This is the only way you can determine if any external power pack is providing the faster recycling time to the flash tube.
First, without the external battery pack. Record the amount of time it takes the red ready light on the SB800 to come back on after you fire a M 1/1 (full power manual) shot).
Next, attach the external power pack again. Turn everything on again and shoot it again in M1/1 . Again record the amount of time it takes the red ready light to come back on. It should be noticeably faster than the 4 internal AA's.