Similarly my instant gut reaction is also a power supply issue. Place sufficient caps as close as possible to the SIM800C , even mod a board to supply a different power supply to the SIM800C to further pinpoint a power problem, scope out the current spikes etc.
Its the transient nature of the power requirements, not the average which causes problems. Although 3A may be enough for normal operations, large current spike requirements can cause voltage sag for transient current requirements and thus cause brown outs. That sag is exacerbated by insufficient track widths and layouts, insufficient decoupling. That 3A capability might be reached under perfect, slow resistive conditions, not taking into consideration heat derating and so on.
Not saying you haven't thought of all these, just throwing out thoughts.