Will a multi-stage charger pollute the output from a 12V battery?
The battery has low impedance the charger has high impedance, the battery will dominate the voltage waveform.
No battery produces a stable 12V,  well a 10S NimH or NiCd battery comes close, possibly close-enough. lithium and lead batteries operate in the 12V ballpark, but do not stay close to 12V over their useful charge level range.