Very late reply here. I received the Xbox logo for Christmas. Basically the same setup, runs on batteries or a USB cable. Like you say, the problem is when it is turned off, it will default to off on power restore and you will need to press the button to turn it on again.
The only work around I can think of is to find a way to keep the button pressed in so once it receives power again it turns on. Maybe some strong tape over the button can do the job? This wil cause the other light modes to not work, but if you just want it on, then no problem.
Apart from this, maybe opening up the light and bypassing the button.