Sounds like a neat system, all the best!!
To localise sounds, you need the time-difference-of-arrivals across channels. If your direct path is the strongest, the normally simple cross-correlations are enough, which can then be used to find the 3d position of the birds. The math required to the positions are in the answers to the question pointed out by @WMXZ.
I wouldn't really bother with temperature, humidity and other such weather measurements. Yes, sure it will affect localisation through the speed of sound, but I've done this broad comparison of with and without detailed weather data and remember at most 1-5% range difference.
The absorption of sound for birdsong type frequencies is almost negligible. For a more detailed reference on weather conditions' effects on sound speed and attenuation check out Goerlitz 2018 (link)