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.

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.

Hello, I am using a T-A7608 and I cannot get the A7608 module to work when it is only powered by the battery. If I plug in the USB cable I get the desired operation with sending SMS. Please note that SMS is sent after waking up by RTC from deep sleep mode. If you had an idea or a code example to try because I've been looking for weeks. THANKS