Line out cannot drive audio isolator
"[radio] passes all kinds of disturbances it picks up from the power supply to its line out."


Trying to "fix" electrical noise after-the-fact is like putting a band-aid on a knife wound... totally ineffective. It is much better to tackle the source of the noise directly.


  "[it] appears that the [radio] line out is not strong enough to drive this [audio isolator]; I lose all low frequencies"...


Forget about the isolator for now. The TDA datasheet states an input impedance of 60kΩ, which is not much load at all; the radio should have no problem driving that directly. I suggest investigating the following:


Check the cleanliness of the power going into the radio. Most good multimeters, on AC volts mode, will show noise up to 100kHz or so. If you see >0.010v AC across the radio's DC supply, then that is a likely source of noise. Of course, an oscilloscope would be the best tool to measure this. Solution: might be able to filter out power supply noise by adding a 0.1uF and 10uF, perhaps even 100 or 1000µF "bypass" capacitor across the power right at the radio.
The TDA datasheet also states a minimum SVRR (supply voltage rejection ratio) of 40dB, which means noise on the TDA power supply can be audible. The solution here is more complicated, as the TDA creates large power transients, and at much higher frequencies, so is very demanding of bypass caps. "Low-ESR" types are pretty much required. The TDA should already have several large bypass caps... if not, add some and measure the AC across them.
If using two different power sources which are not fully isolated, current can flow between their ground leads, resulting in "ground-loop." This current usually results in "hum", but can cause other effects (up to the destruction of one or both devices.) Solution - try to power both from the same source. Easy way to test - power each from batteries.
Long, unshielded cabling between the two can result in electro-magnetic interference (EMI) induction. This is where the wires essentially act as antennas, and pickup radio-frequency (RF) noise from the area. Solution: use shielded wiring for long runs and sensitive inputs. Short runs (hand-width) should not matter, however you are near an RF transmitter (BlueTooth device.)
In section 6.3 Input Impedance and Capacitance (assuming the TDA7492), did you place a 470nF (preferably ceramic or polyester) cap between the radio output and TDA input?