Hi Eric,
3S LIPO is okay for receive. I use 3S LIFE (under 10V) for receive. A voltage of 11V or more is only needed for TX.
I do not see the issue you are reporting. Is there a charger connected to the batteries while you are operating? Those use switching circuits and can introduce noise. Are the cables to you batteries longer than you 12V power supply? Depending on your antenna, the power leads can become part of the antenna and pick up noise. In which part of the spectrum do you see the noise? You can use bandscope in Quisk, SparkSDR and other software to see the entire HF spectrum. You should be able to see a noise difference between with and without batteries. Are there filter or band differences when you run with battery? Most noise comes from 1.8MHz and below. The N2ADR board usually inserts a HPF to block this noise, but if you operate on 160M or do not have your filters configured correctly, then you will see more noise.
73,
Steve