Some progress...
After playing a bit with trimmer I'm now 100% that previous solutions with arduino were not ok.
When the tools circuit gets powered up (after you press trigger button) the tool will pull up COM line to 5V.
And with arduino we should be pulling this line down (to send a signal) with mosfet like in the shematics above. (Mosfet connected to pin 11 of MCU).
I have now connected aliexpress board to my trimmer to check if it works.
Trimmer shuts down after a few seconds, because the signal is wrong for my trimmer:
Comparison:

There are quite some diffrences.
I believe the tool shuts down because it did not understand the last request. The requests that this aliexpres board is sending are much closer to your captured signal than mine.
What I didn't do but should have is to attach all bms lines to 20 resistors to simulate voltages of 20S battery. Will do this in the future but I have doubts that that will help with the signal.
Next on TODO list is to try to recreate a circut to correctly simulate battery communication. (Logic level mosfet connected to ESP32).