The problem is solved.
When operating in battery mode, you must not forget
// disable brownout detector allows you to turn on the simcom module without interrupting the power supply
WRITE_PERI_REG(RTC_CNTL_BROWN_OUT_REG, 0);
because the voltage across the battery drops by 1.5 V which causes a “Brownout detector was triggered” and therefore a reset.
The inrush current when starting the A7608 is around 2A. It might be necessary to add decoupling capacitors on the card. I have the impression that some are planned but not welded.