Would a dying battery trigger the icon though, even if the car is running?

It could do.

The amplifier starting up (when the player comes out of standby) can be a pretty good "shock" to the system -- a sudden request for, say, an extra 25 amps of power. Think of your battery as a reservoir to supply this extra demand until the alternator can take over the load.

If the battery can't do it, then you will get the symptoms you describe.

Try coming out of standby with the engine at 2000 RPM or higher, see if that makes any difference.

tanstaafl.