Hi, BirdNet is a fascinating project. You have me intrgued and I may set up a server soon.
As for fine tuning your automation, since you are already using MQTT, I suggest using MQTT auto discovery message for CommonName, e.g. sensor.birdnet_{{ CommonName }}_lastSeen to set up a sensor to track the last seen timestamp of a bird. In the does not exist, send MQTT to set up, continue. If sensor.birdnet_{{ CommonName }}_lastSeen does exist, use it for cooldown on that CommonName. FInally send MQTT message to update sensor.birdnet_{{ CommonName }}_lastSeen. You can update ignore species to a label on the sensor. The added benefit is that you get a sensor that you can use elsewhere for each CommonName.
The one thing to check is the MQTT auto discovery sensors are not saved to registry so need to be reset on Home Assistant restart. For most cases this should be fine as your automation will start from scratch ofter restart as the sensor will not exist. Looks like the label survives restart (just tested as I have weather sensors via MQTT auto discovery).
Anyhow, I am busy on a few other projects, but if I get time I will have a go at BirdNet and may even be able to test my ideas for the automation.
EDIT: Check here for scripts regarding MQTT and auto discovery.
TIP: I always find it is best that when you want to do X in Home Assistant it is always best to get your checks into a sensor, then life becomes so much easier.