Since you called AppleCare and explained your symptoms, my guess is they cannot run the diagnostics remotely to establish a true failure (or if you agree to pay for the repair) and pre-order the part before you present the Mac for repair in the store.

If the part isn't in stock, Apple should be able to quote you a delivery time to order the part, and discuss if you want to leave the Mac for service, choose mail in service or leave with your mac and return later to get the repair done as same- or next-day service once the part and the Mac are in the shop together. Of course, your wait time might be longer if no one is free to start work immediately or the part needs to be retrieved from a nearby stock room. This assumes the technician has all the parts, adhesives, solvents in place and has done a dozen or so of this exact model so they only have to refresh the steps and not have to carefully study the manual and find each screw and piece to remove. When this question was asked, it was nearly 100% yes - all computer batteries from Apple (including the newer unibody models where the battery is not consumer replaceable) are easily swapped in a 10 to 35 minute procedure to open, inspect, replace, test and document the repair.

Many batteries are glued to the top case, so the batteries may or may not be swapped locally or sent out to a repair center with larger staff and parts warehoused outside the retail store.

Genius bar macbook