The issue is that she *is* American, whether or not you choose to get that passport or register her. So no, she can't just be British because the citizenship has already been passed on, even if you don't notify the Consulate.
This presents a number of logistical issues you need to be aware of. One is, of course, the double taxation; the other is the fact that Americans must, by law, enter and leave the US on a US passport. These are issues you will have until she is old enough to renounce her citizenship on her own, as that's not something you can do on her behalf.
Sucks, but that's the reality of dual nationality with the US.