No, the Target exclusive AT-DP with the pack-in driver is not in circulation yet. The counts you are seeing are for the first wave of vehicles most likely. And yeah, the AT-DP seems to sell out fast.
I've seen indications that the AT-DP with Driver is actually in some of my area Target stores. I was able to come by that info from the Target website and searched for the exclusive AT-DP in my area.
That feature doesn't really work on Target's site because it treats the exclusive vehicles the same as the non-exclusive one. So basically, if Target.com says your store has it, they probably just have an Anakin Jedi Starfighter or maybe a Phantom with Kanan or TIE with Inquisitor. In essence, it's a crap shoot.