The OP seemed to ask in the original post if bypassing the AC to DC PS in his equipment and connecting a battery would benefit his playback.
A battery power supply converting 12 volt DC to 120 volt AC is not the same topic.
Ron Sutherland use to use battery power for his phono stages. He says he gets better performance with 120 volt AC power supplies. I don't see many equipment manufacturer use battery. Some do. Dartzeel on the preamp. My Lino Channel D phono pre. They are used. I don't know of any amps that use them.
Batteries are current limiting. Lithium less so than led acid. A deep cycle lead acid is very slow to release power.
There are so many variables you can not make a carper statement one is better than the other. You could modify a piece of equipment and make a judgement on that one piece.
As to wall power vs a battery, most people have crap wall power. Even those that add circuits usually have some issues. An isolation transformer is also a benefit. But you need a good one. Not some deal you scored off ebay.
Getting your electrical to a premium level is money. So is a true battery system providing 120 volt to a rack. I calculate it to about $18k in parts if your going to drive a whole home system. The battery are the lion share. The inverter is a critical link.
And after all that, you still have not addressed RF that is entering through the equipment itself.
FWIW. The best inverters I have seen have about 3%thd. Depends on the load to some extent. The worst houses I have measured have about 5%thd. Most are about 3%.
Low frequency inverters give better peak power than high frequency. Never use a modified sine wave inverter.
Remember, lithium batteries do have a tendency to burst into a fire ball durring charging. Don't buy some cheap piece of crap and use it in your house. Buy a good one. And get the $500 charger they want to sell you that is made for that battery.
You can also just buy a Tesla powerwall and plug directly into it. Or a Stromtank. Or a PS Audio. Or a Pure Power. Funny how so many people have owned and sold that gear because they like the wall better. And yet audio forum members who may or may not have any relevant understanding of electrical components are going to cobble together some parts and outclass a full scale corporation. Not saying it cant be done. Just saying, real business are trying and partially succeeding.