The fan inside the B300K is so quiet, i only hear it when i power on the battery pack. I do have a B210 which have also a fan. If i wouldnt now it has one, i wouldnt even notice.
You can expect a near fanless operation for what i notice so far.
I have done some testing with the B300K battery, including with my AC300 system that powers part of my home full-time. The only time I have ever heard the fan run is when the battery is powered up. Even with full solar charging (around 2,200+ watts) I have not heard it come on, or when the 1,100 running watts water pump is going. Now, I do have 4 batteries, so the power is split between them, and the system is in my basement which is decently cool, probably somewhat cooler than 20° C, but not by a lot.
I will say that the fan is considerably quieter than the AC300’s fan when that is running, and on my system the AC300 fan does run periodically.
I confirm your tests: the fan is only audible for 2 seconds at start-up.
By cycling the battery reasonably the fan did not trigger. I recharged the battery at around 700/800 watts, it finished at 32°C in a room at 23°C (information checked in the secret menu of my AC 300).