I know nothing about the wiring in that Cooper, but those connection points don't sound like the proper places to be hooking up the power lines.

First, do make sure to run the player (and its amplifier, I assume you have one) from a line that's run straight from the battery. If it's feeding an amplifier, it needs to be fairly heavy-gauge wire, probably heavier than the cigarette lighter's wire. Make sure you fuse it close to the battery somewhere. This might solve the problem.

If that doesn't solve your problem, then you're looking at power issues with the car in general. I don't know how well-regulated the voltage in those old Coopers are supposed to be, but I'm guessing it's not supposed to drop below 10 volts, which is what that "battery symbol" is indicating. Perhaps the ol' Coop' needs a new alternator, a new battery, or a new voltage regulator, and your Empeg is just telling you about it?

Note that your ignition wire kluge, although not optimal, has nothing to do with the voltage drop and the "battery symbol" appearing. Your problem is only with the constant-power line. If it had anything to do with the ignition line, the player would fully reboot rather than show that low-voltage indicator.