Does the Apple Genius Bar replace Macbook Air battery on site?
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. 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. 

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. 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.

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.

My experience is each store has sufficient stock to do several repairs of a battery type, but that being said, I've also gone in on a busy day where they had three machines needing the same part as I and I was the third in line and therefore had the option of leaving the Mac or waiting for the part to arrive.

Worst case, you get a diagnosis and don't leave the machine in for repair, but have options to pre-order the part when you return or find another store. If the store mails in your Mac for repair, often it can get overnighted to a return location of your choosing and clearly wherever they ship the machine will be same day service for an in-stock battery at the main repair depot.