I've figured a workaround. I hope this helps everyone who had the same problem as me with a broken original SSD in the MacBook. What I used is another MacBook and an external SSD.
Connect the external SSD to the second macbook. Download the full Catalina installer from Apple. Double click and select the SSD as drive. Press install and wait until the macbook restarts. Hold the alt button and boot in original drive. When booted disconnect the SSD.
Connect the SSD to your macbook you wish to upgrade the firmware. Hold alt and let it boot on the SSD. The install will fail at the end but that is ok. Shut the macbook down, connect your new internal SSD and boot. Start internet recovery and guess what: your new SSD works!