Does replacing an audio device's AC adapter with a 9v battery require filtering?
I would say that from the point of view of noise, spikes, and in general EMI, batteries are a bless, because they look like the ideal DC source. They have an internal resistance that is quite low, so they can even provide current even in case of a pulsed peaky current request.
They have also some stray capacitance so in some way they are already a &quot;filter&quot;.
They are also insulated from ground and can avoid much of the common mode disturbance. Of course a large battery has some stray capacitance to ground, but it is not your case, probably; if so, you can prop up on spacers and separate from earthed parts and ground potential.
What they don't do is providing a stabilized output:

if there is a significant current absorption, you will see a corresponding voltage reduction due to the internal voltage drop;
with use, when the charge reduces, the voltage is also somewhat reduced, following the known discharge curve typical of each type of battery (e.g. LiIon, NiCd, NiMH, etc.)
when you connect a charger and you keep everything connected (so you are still supplying your device), be careful that the voltage is higher and may be above the rating of the connected device.