Electrical – “Low Pulse Trigger” and how to use it

You need a capacitor large enough to hold enough energy to reset the relay when the +12 disappears. Minimum pulse width is stated as 30ms. Assuming a 1V drop would be okay and allowing for say 45ms time, you'd need a capacitor C = \$\frac{t \cdot I}{\Delta V}\$ or 0.092A * 0.045s/1V = 4,000uF. So maybe 4700uF/16V.

Then you need a control circuit to detect the power loss and pulse the set and reset coils. It may be possible to do an adequate job in your case with a few passive parts and a 2N6028 but if you need reliable operation in more pathological situations such as slow brown-out of the 12V line you might want to consider more sophisticated circuitry or even a microcontroller.

The problem is the diodes on the audio path, it simply does not work if there is even one diode on the audio path, since audio is AC signal.

Simply arrange the DPDT relays to either bypass the audio, or route it via the send/return connections. You should use both relay poles for this so you can't drive the LEDs with relay, but there might be ways to drive the LEDs with other ways from the foot pedal switches.