SIM868 powers down after few seconds
It is stated in the datasheet that the module will shut down in a few seconds if the supply voltage drops below 3V. 



The use of two diodes in series for dropping the voltage is not optimal since, their voltage drop is dependent on the current flowing through them. Considering their smallest voltage drop (2 x 200mV = 400mV), would would supply too much voltage (4.6V) to the module. Considering their highest voltage drop you have two scenarios:


With data mode GSM850 there's a current consumption of about 378mA, which causes a voltage drop of 2x300mV = 600mV. Hence, the supply voltage of the module is about 4.4V.
During Tx burst there's a 2A peak current. You would have to measure how much the voltage drop is with the aid of an osci. I reckon that is the problem which is causing your module to reset.


You can try adding more capacitance to the VCC pin of your module in order to prevent this voltage drop. However, the problem with the imprecise supply voltage still remains. If you are using the 5V only to bias the IC charger, you could also modify the feedback of the IC MP1484 so that the output voltage is 4.4V, so that you wouldn't need to add the two diodes.