just to add 2 cents,
the 12 volt line in cars is dirty as all hell in old cars. couple that with questionable ignition wires and you are looking at a bad day.
look at applying a noise filter for the power for your stereo (usually a common mode choke coil or two and some caps.) this will eliminate a bunch of noise.
check your alternator output and see if you have a pile of AC coming out (indicating a failing bridge rectifier) this is often a common source of noise.
also look at resistor plugs and wires to reduce noise from the ignition system. NKG has a pile of OEM compatible conversions.
except this is happening without the engine running.. so none of that applies. DC voltage from a battery is steady and even if your input ground is directly tied to it, will not change the signal in the slightest. With the alternator charging the battery, it will get bumpy and that, along with spikes from relay coils discharging and other noise will cause some issues when the path for the signal is not isolated from the electrical system.
Furthermore, a chassis is one big wire, but it is also a ground plane and a positive wire running through the car becomes an antenna to pick up noise from the spark plug wires and any other devices that emit EMF.
So while you are right that a car's electrical system is noisy, a ground loop will only be an issue while the engine is running.
In this case I think we have a voltage differential due to a common ground in the BT device that is feeding back through the makeshift input. Usually this would just cause a pop as the input was plugged in and the voltage spiked, but my guess is it is big enough that once the loop is completed the DA converter or PA circuit in the Delco is going crazy due to the voltage and causing the feedback.
Other things that come to mind that would cause this would be the input polarity being backwards (so the ground from the BT is feeding the voltage differential into the positive input on the Delco and causing a loop), or possibly just a straight up lack of ground on the DA circuit so it is getting it's power through the ground on the input wire. But either isolating the input common lead with an isolator circuit or just grounding it to chassis should stop it.. if the polarity is backwards on the input then an isolator would result in no sound at all (incomplete circuit).
@Howard, I'm not sure we are on the same page. The Delco unit is grounded through its chassis to the chassis of the vehicle (most likely), and this is to power the unit. But I am not talking about unit power here, I'm talking about the line level input. Take either the outside of the RCA jacks or in the case of a 3.5mm plug, the shield wire in the cable itself (the bottom part of the jack), and tie it to ground (the radio chassis) after the connection between the BT and Delco is made. This is an old stereo trick we used to use when there was a voltage differential between the source and the amplifier circuit. If you are already grounding the line level input common side to chassis, then this is not the cause (obviously). In fact, it might be that the BT unit has a floating ground (not the same ground as the power input uses) and this is why the unit freaks out and sends crazy feedback. In that case I would add a ground loop isolator to the input to prevent the ground on the Delco from affecting the output of the BT. The power side of both units is seldom a factor, regardless of where you ground. As long as 10-15 volts goes in, you will get ~5 volts out on the adapter and the unit will power up. If you had noise while the alternator is going, I might say to play with different ground sources. If the unit was cutting out (motorboating) when you turned it up, I would say the ground is the issue (low voltage due to bad chassis ground at the battery). But getting feedback while just on battery power indicates some kind of incompatibility between the output of the BT and the input of the Delco. Hope that makes sense. Not saying there is a solution here, just a couple things to try. Good luck!