Hi @felmue and many thanks: we'll have look.
From our side, good news. M5Stack guys gave us the tip to set ECHO OFF once the module is resumed from the sleep, so soon after PWRKEY sequence, our code checks module status and then set echo off.
In this way we're able to launch the MQTT client (provided with the new IoT Base CAT-M palette) and publish a topic.
We were not able to catch the "EXIT PSM" URC. We noted however that if the code invoked earlier the "network active" brick the URC shows +APP PDP DEACTIVE (we guess that TCP sockets are closed as well as any protocol layer above IP is terminated) while if the code never invoked "network active" before the PSM then we catch "ENTER PSM" URC.
At this stage our investigation around PSM is pretty satisfactory of the results achieved.
We'll keep you updated about any further progress.