So, i can't see under the chip for what pin goes where and I'm not at home to test points...but it appears as though they are using multiple pins...granted, three of them would have to be devoted to push pin. it does look like the pins are devoted to the blatantly obvious components. Unless they are treating the din pin as both input and output. (possible?) if it is an input then the current draw check would be logical from there. anyway...It would be cool to pull current draw. 
No it's not possible to use that pin as any kind of output. The chip just doesn't have that function. It would be documented in the data sheet, and there would be some method of triggering the function to output data, and there's nothing like that.
I think you would need to measure current draw on the power pins, not the signal line, because that's just carrying binary data, that you are sending, so anything you measure there would be a result of what you sent, not what's happening on the light string, and that's if reading the current of a data line was meaningful anyway. It would be interesting to know if the resistors are connected to some kind of current sensor. Can you take a high quality photo of the boards? I couldn't really tell anything from the previous photos.