EDIT: You should be able to connect the play and charge kit directly to the 360 and the cable should sync the controller to that console right away. I went for the 2 workarounds because I was guessing this wasn't working for some reason but I just realized you might not have tried it yet (and trying it with the controler connected to another usb power source).
You can bridge 2 AAA batteries with a few copper coins to keep the pressure (pennies will do, or any brown colored coin). Keep in mind the coins will be conducting electricity and as such might give you a very small jolt if touched while the controller is on. Nothing dangerous, similar to what you get if you lick a battery.
A video of the coin methode
The alternative, is drilling a hole directly leading to where the button would be, and then use a needle to push the sync button. Using powertools should always be supervised by a grown up, so if you are a minor, just ask your mother or father for help.