Jonathan Goldsmith

You can't really see it in these pictures, but there is a small hole at one end of the "pull tab". This is just the right size hole for the pointy end of the apple tool spudger, and thus is very handy for pulling the tab out.

Okay, the problem would seem to have been with the screwdriver that came with my iFixit battery "Fix kit". Tried another screwdriver, and no problems at all.

Now my problem, however, is that the screws on this bracket do not seem to want to unscrew. :(

Pan is right. Not removing the battery means you will run the risk of shorting some circuits not meant to be connected, and thus can damage those components. This can happen very easily, and without you even knowing it. Additionally, the internal components of a digital device are not meant to be disconnected while having any amount of power applied(and even though your device may not even turn on because the battery is completely shot, there very likely will still be a small amount of power coming from the battery to the internal components of the device). Doing so will likely cause an overcurrent condition that will damage the components of the device.