I would recommend that you get a ground loop isolator.
Thing is the typical ones aren't designed for your setup.
You'd need a high level converter then a ground loop isolator.
They do make a isolator that's built into a high level adapter...
It's worth the try...
As for what the other guy said... He's right...
Typically you should never ground ground something to the battery. It's a no no... But sometimes...
Alternately though... I've NEVER EVER grounded anything to a battery because you never ever need to. You can ground to the location that the battery ground strap/wire goes to. So follow the negative strap/wire off the battery to the chassis remove that bolt and use a properly sized ring to ground your ground wire... I'm guessing this will solve your issue. If it does not then there are other options that might get a bit complicated but they are doable.
For the ring... Use something decent made of copper. Load it with solder... So if it's a crush then crush it then load it with solder.
I don't know the size of the bolt but let's say to get a decent ring termination it ends up being for 8 guage and your using 12 or 14 guage you can strip back extra wire then fold the fire before inserting it into the termination. But like I said please please load it with solder. You'll push the tip of the iron into the wire wait a moment and then start shoving the solder into the iron tip then just freely into the wire and termination. If you don't do this there is a good chance that eventually the termination fails one way or another.
Another option would be IF you wanted to improve audio beyond just the sub I'd recommend a Rockford Fosgate DSR1 it's going to cost you $220-300 but it's like having a $2000 DSP!!! Ive spent thousands on DSPs never has truly top notch quality and integration been so readily available and affordable.
All you need is a DSR1 and a small inexpensive hideable 4CH amp or two. Id only do this if you want truly top notch audio and plan on replacing your front speakers.
A 4CH would do your front speakers I'd recommend disconnecting the center in this situation. Youd buy the tweeters and lows separately without crossovers in this scenario.
The crossover would be the DSP after using a microphone to calibrate the system which is where a DSP shines. You don't need the best speakers in the world. You just need decent ones and decent small digital class amplifiers.
But that's just if that's your direction...
If you just wanted to go the least expensive route then so what I mentioned first and eventually if it's not the sony system then disconnect the center, add bass blockers to the 6.5" speakers (bass blockers are dirt cheap and increase the available amplifier power that can be sent to usable frequencies by blocking the lowest frequencies that the speakers can't play anyways).
You should def disconnect the center. If you do disconnect the center never run the car in driver audio mode. That uses a DSP that expects the center to be present.
Anyways hope this stuff helps
20 years of absolute audiophile level car, home and theater level experience, 20 years home and vehicle automation/security experience. 20 years of calibration experience (DSP, Time Alignment, making **** speakers sound better). 7 years of active cross-over experience (using electronic DSPs to REMOVE the passive crossover networks found in speakers including my first setup martin logan electrostatic speaker with the entire crossover network gutted)