Well crap. I just read a few places that AlienBees strobes have bad luck with modified sine wave inverters, which is what that Black & Decker unit is.
UGH.
What now?!
Go back and read what I wrote about going to a higher end retailer. When I messed with inverters it was on my sailboat. Give them your specs, bring a pickup truck and a credit card.
That said, I'm trying to figure out why a strobe can't use a modified sine wave inverter. The one you found is only 2X to small but more will allow that kind f overload for a few seconds