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.
This is what it looks like inside (you can remove the 8 screws, 4 of which covered by rubber feet), and it comes right apart.
I guess you could put an ESP in there to both read/passthrough the button input and control the driver board (with some clever coding you can track the current “state” of the light by how many times the button has been pressed, and and expose all modes by simulating repeated button presses), or you could replace the PCB entirely, especially with a board that supports PWM, and drive the LEDs directly.
Keep in mind:
Those are bare diodes (about 2.5 V forward drop voltage), so you’ll need to use resistors to regulate the current
The USB input is rated for 500 mA maximum, and comes with an inline fuse you can choose to keep or remove obviously.
Don’t have the time to do this or a soldering iron with me right now unfortunately, but I’ll follow up in case I end up doing this