For all the fellow Verizon customers, I just got off the phone with them. According to the rep I spoke to, the promotion is good on any Series 3 LTE Watch activated by September 30th. Kind of a short window, considering the first ones don't ship until the 22nd. If you have a 2-3 week or longer wait time, you're probably SOL.
Then again, most reps have no idea what they're talking about, but this one had to ask her superior so I'm assuming the information is correct.
EDIT: Also, it seems the LTE function, at least on Verizon, requires an iPhone 6 or newer due to the need for VoLTE. Sorry 5s users.