Replacing power transformer with SMPS to charge SLA battery
To fully charge a 12 V Lead-Acid battery you need more than 12 V, you actually need about 13.6 V. So you will not manage with the 12V power brick (SMPS) alone.

You could increase that 12 V a little by adding a boost converter set to 13.6 V. Do add a series resistor though to limit the charge current when the battery has a low voltage.

Also check that the power ratings are OK, that you do not load the power brick more than it can handle.

I would not worry too much about the SMPS interfering with the audio as most SMPSes will switch at a frequency outside the audio range. But if needed you could add decoupling capacitors on the supply lines and / or a ferrite core around the wires. But I doubt that is needed here.

Regarding charge management, that's the resistor I mentioned. If you make sure the highest voltage you charge from is 13.6 V via a resistor, the battery will be fine. Lead Acid batteries are far more robust and less picky than Lithium based batteries for example.