It seems to me that IF the battery couldn't handle the radio THEN this would cause the system voltage to drop. After all there are TONS of cars with a Mach 1000 sound system using stock alternator and battery. This would imply that there's some problem with your car's charging system.
Regardless why guess? Test! Here's some information on how to test the car's charging system.
Howto perform charging system voltage drop test
https://www.stangnet.com/mustang-fo...perform-charging-system-voltage-drop-test.56/
Feel good saying for the day. "There is no elevator to success. You have to take the steps". That includes reading the information and performing tests needed to rule in or out possible causes. Those of us out here in Internet land can't help you unless you are prepared to perform some trouble shooting. Otherwise the only way for you to get a solution is by pure chance to happen to run across someone with the EXACT same problem.
If this is the case, I can't help so will need to drop off this thread.