I don't believe brake lights would have anything to do with the battery light.
The battery light will come on for two reasons. It's either overcharging or not charging at all. If it isn't charging, it'll keep running until the battery dies then you're scewed. My first guess would be the common broken wire but you should check other things first...
First and easiest thing to check is make sure you still have a belt... If you're belt broke, you're gonna have a bad time.
If you still have a belt, the next thing I'd do is finger out if you're battery is being charged or not when the engine is running. Start it up and throw a volt meter on the battery. It should read between 13.5 and 14.5 volts(when engine is running).
If it's reading closer to 15Volts then you probably have a broken somewhere in the wiring harness between the alternator and ecu.
The most common place for them to break is in the wiring harness along the top of the intake manifold. Usually near where the harness is zip tied from the factoria. Wires vibrate and chafe through, then hang on by a thread. One way to check if this is your problem(not a sure fire way) have someone sit in the car(running) and wiggle the wiring harness around. If light flickers or turns on and stays on then that's probably your problem.
Easy cheap fix. Find broken wire then repair it. Might have to splice in a little chunk oh' wire.
If it's under/not charging I'd still take a look at the wiring harness. That could be the problem too. I've yet to see a bad Mirage alternator but that's not to say they can't go bad. I'd check the wiring first before throwing a new or even a used one at it.
This post is probably way longer than it needs to be.
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View my fuel log 2014 Mirage SE wussie cvt edition. 1.2 automatic: 37.7 mpg (US) ... 16.0 km/L ... 6.2 L/100 km ... 45.3 mpg (Imp)