Thanks for your fast replies, and possible solutions!
Good to know that this slow voltage drop can lead to failure or damage!
I did a little research on brown-out detection and as far as I understood it should solve any problems if I changed the BOD level to 4.3V.
To accomplish that I could use a second arduino (that I might need anyway), or I could burn the adjusted bootloader to a separate ATmega328P chip on a breadboard and then replace it with the one on my arduino.
My main sources: https://www.arduino.cc/en/Tutorial/ArduinoISP
Arduino Uno brown-out detection