3.3 V Pro Micro clones flashing LEDs after 1200 bit/s touch
I was finally able to solve this by using one of the Pro Micros to re-burn the bootloader of the other one.

For reference, the process is like this:


Flash the example sketch "ArduinoISP" into the first Pro Micro
Connect the two Pro Micros like this:

Programmer (on USB)   Target
GND ----------------- GND
VCC ----------------- VCC
15 ------------------ 15
14 ------------------ 14
16 ------------------ 16
10 ------------------ RST

Choose "Arduino as ISP" as Programmer
Because both are Pro Micros, selecting a different board is not necessary.
Click "Burn Bootloader"