With 8VAC on the " ~ " terminals you should have around 7VDC from "+ and - " terminals, sure your connections are right? If you're putting pulsating DC on the LM393 input it may not work as expected, you could add a filter cap but that would kick your DCV up to about 10.2, you could still use a voltage divider to get down to 9.5.
Also you need a kickback diode across that relay coil to protect the MOSFET.
You will need to half wave rectify your 8Vac not full wave with the bridge you are using, the bridge is shorting the alternator out on each negative cycle.
Why do you need to replace this part of your car?
Are you sure you know exactly how it works?
That wasn't necessary , although it does help to stablize the voltage.
All you needed to do was change the comparator input voltage divider resistor values.
Robert