Hi,
The benefit of a switching power supply is, that you don´t need high value capacitors.
Often - if they are used at the SMPS output - they just cause problems. As you see. Do you see a benefit in them?
If you just need an energy reservoir, then I recommend to put a big capacitor at the input side of the SMPS.
Calculation: 3000uF charged from 0V to 4V within 10ms needs an average current of 1.2A. This may be more than the modem will draw.
If disconnecting the modem makes it work, then you are at the edge of safe operation.
I assume any change of some system parameters (battery age, battery type, temperature...) can lead to malfunction.
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And you say that the power supply output resistance is 1 ohms. Why is this?
(this often is combined with power loss = heating = low efficiency)
If this was my project I´d reduce the capacitor values and improve the SMPS.
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But for sure I recognize that you wrote you can´t change this.
Klaus