Found some time again. Could not solve the issue with disabling Wifi. Further, I got a bunch of ESP32, which should have a dedicated core for Wifi, and I still cannot get it working.
I have recorded the traces with a logic analyzer, compared to a Pro Mini and after the initial pulse, the timings seem to be vastly different. I only have sigrok pulseview available, and I find it hard do do any further analysis with it (including exporting it into any sensible format to import it to e.g. MATLAB ...)
Here are the Sigrok session files for a Wemos LOLIN32 and an Arduino Pro Mini, if anybody is interested.
I have also scoped the waveforms, they are pretty fine in both cases. The Pro Mini is on the same breadboard and powered by the LOLIN, they both have the same code fresh from Github (only change is ATSHA pin to pin 4) and the only thing I did between the two recordings is plug a jumper cable from pin 4 of one device to pin 4 for the other.