When you go to install the picaxe you should put the chip right above the pins for the right trigger. Theres a small slot in the top part of the controller case, which provides a big enough spot to fit the picaxe and the wiring into without removing the motors. Theres two ways to do this. The first way is less invasive, but the second way gives you much more space to use to fit bigger chips(18m2, 20x2) and a hole for the programming jack I've attached some images.
Also I have thought of another way to have a programming jack in the controller without cutting a hole in the controller's case. There is an attachment for the controller called a "chatpad". It connects to the controller using a headphone jack, and two flat plastic pieces with metal contacts on them. If you remove the connector from the "chatpad" you could wire a 9 pin to the jack(the gray piece in the picture below), making a special serial download cable. You would then cut the traces in the controller that would connect to the chatpad, and wire them up to the picaxe. this could also be done with a play and charge kit but it would be more difficult to make the cable.