Li-Ion battery for RGB strip
You haven't looked hard enough if you didn't find a 3.7V->12V step-up for >=1A. 

I've built stronger things based on SMPS controller IC reference design circuits years ago. Since this is a design, and not a shopping recommendation site:

Try going to www.ti.com, use the (mobile) webbench power supply designer (it's on the home page), and design a 3.7 V -> 12 V converter. It works.

Also note that it's quite possible that you'd not want to use just any LED strip in this case – after all, they often contain series resistors themselves, which just wastes the voltage between the actual forward voltage of the LEDs and 12V. Meh.

With a bit of thought (generally: take a step-up controller, and instead of using a voltage divider to give the feedback voltage to achieve the a fixed output voltage, use a shunt resistor to achieve a fixed output current), you can design an actual constant-current LED driver instead of a constant-voltage voltage source, and get rid of the series resistors. 

Also note that, depending on the currents, bijunction transistors as used in your block diagram (it's really not a schematic) might be suboptimal and you might want to go for MOSFETs, or just switch the step-up on and off (which requires controlling of some ENABLE pin or some secondary comparator, instead of actually low-side switching, and thus might be more effective and easier).