Have you tried swapping in a gel cell or SLA battery just to see if the noise goes away? It seems like a simple test to isolate the cause. I don't think the 14.6V would harm the battery if you just did it long enough to check for noise.
ETA are you connecting the radio in parallel to the battery and charger so the radio is taking the full 14.6V charger output?
I am curious how all this works because I have a 12000mAh LiFePO4 battery I want to build into a small plastic ammo can and equip it with a solar charger. But, I'm not planning on running the radio while the charger is connected, but I probably will be using USB ports to charge other devices from it.
Ideally the charger would fit inside the ammo box along with the battery and possibly a 110V charger as well.
For my intended use, noise is a secondary consideration to price.