You answered your own question.
Your battery is low or maybe old . (you see lots of 06 batterys reported as bad)
Plus if you drive only short distances and let the car sit a lot this also
depletes the battery.
When you do an install without disconnection of the battery the car stays
awake which is a considerable current draw on the battery.
The IBS (INtelligent Battery System ) detects this and starts to turn
stuff off to save the charge .
Get a battery tender such as a Ctek .
When you start to approach 2000 watts in add on equipment you may have
a problem with the charging system keeping up with load.
Also is your IBS Cable still plugged in at the battery distribution
strip on top of the battery?
I have heard of people unpugging the IBS Cable to get the car started I wonder if it would defeat the IBS if you unplugged it
allowing the stereo to run.
I would measure your battery voltage first.
You should be able to hear the technic harness relay turn
on when you unlock the car and see the power led's on your
amp go on.