Birthday (and other) buttons are available in lots of places although sometimes they run out, so just keep asking until you get one. We have gotten them at character meals, from cashiers in the stores, and even from one of the CMs who walks around the parks handing out maps and times guides. Other places that might have them are the guest services on Buena Vista Street in DCA, the info/times board kiosks, and World of Disney.
BTMRR has fastpass machines. I have heard that they are sometimes not used on less busy weekdays.