I've been reading about this a ton over the last month also.
Essentially your bullet only cares about Air Density.
Air Density is a combination of Temperature + Station Pressure (Absolute Pressure) + Humidity.
That's why you can ignore Barometric Pressure and Elevation. Neither actually affect your bullet's trajectory because neither are factored into your actual Air Density.
Many people just leave their Humidity set to 50% in their ballistics software because it has the least effect on bullet trajectory. Temperature is easy to see from any device and input that data. That leaves us with the confusing area of Air Pressure/Baro Pressure/Altitude/Etc.
If you have a device that measures Station Pressure (which you do), just use that in your ballistics software. Some software will actually grey out the Baro and Altitude fields if you input your Station Pressure. But not all do so you'll need to pay attention to that.
If you didn't have a device that measures Station Pressure, you would need to input the Barometric Pressure and Altitude from known sources and the ballistics software would estimate your Station Pressure from that data anyways.
For shooting, use Station Pressure measurement from the Kestrel.
Now, if you're hiking up a mountain and wanted to know your elevation changes, you could use the Kestrel for that if you calibrated it with Barometric Pressure first. In this scenario you don't care about your Station Pressure, you care about your elevation changes.
Clear as mud?
I was confused about this for a long time. Then once you figure it out you feel pretty silly. Because it's not complicated at all.