Power supply - Mega 2560 / SIM 808


I'm using a Mega 2560 and SIM 808 module.
Based on the data sheet SIM 808 will need up to 2 A peek.
(Wiring attached)

If I use separate power supply (9V 1A) for Mega & SIM, everything is fine, working flawless.
Sadly, I don't wanna use 2 power supply for my setup.
What I tried so far:

  1. Power the Mega and forward power to SIM (based on spec this should be 5V 0.7A) => SIM turns of after ~3 sec.
  2. Power the SIM and forward power to Mega => Everything works fine, except I'm not able to communicate with SIM anymore, only timeouts. Maybe this disturb RX / TX somehow? idk
  3. Using a breadboard power supply, and forward to Mega/SIM separate => SIM turns of after 3sec, as I wasn't able to find a breadboard providing more than 700mA.

What I will try:
A) Buying a SIM 808 shield, but no clue why this should work, while 1) isn't.
B) Will try DC/DC Converter XL4015, but even this will work, I have to destroy my adapter, which isn't a safe thing to do.

Anyone of you have a clue?
How do you do connect / supply your SIM808 (or 800L/900, will have the same 2A peek based on datasheet)?
Somebody tried to used this with a battery pack while connecting to an internet page? Will the battery pack enough for both devices?