DC/DC boost converter voltage drop when connected to load
It sounds like that one or more of your components are not rated for the input current your stepup requires. You need to look closer at your battery and your inductor. You are roughly triple your output voltage of 12V, that means your input current will be about 3 times the output current: 800mA@12V = 2.5A@4V. Will your battery supply this current? Will it also supply the double amount if the switcher is drawing the current in pulse mode? Same for your inductor, is it rated for at least 5 Amps?