Help me hack this other speaker: Where would I introduce two different DC voltages?
If you don't know enough about linear and switching power supplies to recognize what you're working with, then you probably got lucky the first time.  Or you followed someone else's instructions for that exact design who did know what they were doing or got lucky themselves.

In general, some things to watch out for:


Many topologies can handle being backdriven when powered off themselves, but not all of them.  If you find the right place to hook up your battery, the AC supply will most likely do nothing, but in some cases it may actually present a short-circuit to your battery.  This could be because it uses an inductor to eliminate most of the switching noise from the output, and at least one of the switching transistors themselves defaults to "on" (some types do that, or perhaps the battery voltage gets around enough to turn it on).  Inductors pass DC like a short-circuit, and the stuck-on transistor completes the short across the battery.  Or if it's a linear supply, the regulator will probably NOT like having its output powered directly.
Multiple power rails are by no means guaranteed to be connected.  They could be, if the primary supply produces a relatively high voltage which then feeds one or more regulators, but more likely is that there is a single oscillator (which could be the AC input directly or a higher-frequency clock) that drives a single transformer that produces all of the output voltages in raw form.  Each output then has its own post-processing, so to speak, but you can't feed DC back through the transformer.  If your unit does any sort of active signal processing besides amplification, then it probably does have multiple internal supplies.  (volume, balance, and simple bass/treble can be passive, but that's about it, and even those functions could be active anyway)
Some of the capacitors may not be used for the power supply, but to AC-couple the signal from one module to another.  One example of this might be to offset it from being centered on 0V to being centered between the internal supply rails.  The voltage across that cap would then be equal to 1/2 the supply voltage.  Connecting a battery to that one will not power the circuit and might make it clip (distort) easily because you're forcing the signal to sit too close to one rail or the other.  Or if the cap is used for something else, then the symptoms might be different.


The way I would approach this is to reverse-engineer the circuit under a bright light so I could see the copper tracks, starting at the AC input.  Do some research on switching power supply topologies and see if you can match one up with what you have.  By that time, you might have learned enough to determine if you do indeed have multiple supplies and if you need to unsolder something to isolate it or just leave it in there as a diode-OR.