I was quite desperate, but I solved the problem myself. It was a malfunctioning cable...
The jack connector was faulty (the second one I tried also had a problem), the cable was broken, and the car charger plug was also faulty. I replaced the entire cable with all the plugs, and the problem was solved.
It just goes to show: use perfect cables. The Anker and the AM3N are working perfectly now: no more alerts and no more low-power notifications.
Kudos also to the ZWO software, which displays notifications in the event of a power problem. Perhaps a good idea: add more information about power issues in the AM3N manual and provide a comprehensive description of the beeps the mount can make.
Case closed.