Amp clickin in and out
As long as there is power to the Remote terminal, the amp is on.
If the rest of the system is off, the amp probably won't suffer, but why take a chance. Also, it will drain your battery overnight.
Put a switch in there somewhere. Make sure you can cut the power.
Great job on the click fix.
This does change my view of your original problem though.
Your description covered every symptom of power starvation.
The Remote is the amps 'On' switch.
Put +12v to the Remote, the amp turns on.
It's how the head communicates with the amp.
If the head units Remote Out was experiencing either an intermittant connection to the amp (loose wire, intermittant short) or power fluxuations of it's own, that would produce symptoms similar to power starvation (clicking off, coming right back on). Only it would be random.
Your Bass vs. Volume observations would also have been random. They weren't. You were able to reproduce the problem at will.
You probably have a multimeter handy.
Since running a hot lead to the remote terminal on your amp stopped the constant 'restarting', the next step is checking the Remote wire (condition and voltage) going from the head to the amp.
Start at the amp. Connect the negative lead to a ground. Connect the positive lead to the remote wire coming from the head unit. I assume you disconnected it from the amp. It shouldn't matter.
When the head is on, it should be +12v.
With the meter still connected, slowly turn up the bass.
Then the volume. The voltage should remain constant no matter what.
This has been mostly FYI.
Rule #1: If it ain't broke, don't fixit.
Feel free to contact me with any other questions, problems or just to show off.