Go with the left one, "olde skoole". To calculate your cut off voltage, take the forward voltage drop of your forward bias diode and take it away from your desired cut off point. You then select a zener with this breakdown voltage.
You can do this, as far as I can see. But then you have the risk of the cap discharging into the load or the coil, drawing the power away from it's job of ensuring a smooth base input to the Darlington. Of course, that may be wrong, either way probably works.
Also, ensure your cap is rated for twelve volts.
You mentioned that you had a PSU, what is the minimum that your system runs down to? Allowing discharge of your batteries while the system is not working is inefficient, the batteries are being drained for no reason, additionally lithium ions prefer shallower discharges. See
here. If we take the depth of discharge percentage and multiply it by the cycle life, we can total lifetime discharge capacity:
100% * 300 ~ 500 = 300 ~ 500 LDC (low)
50% * 1200 ~ 1500 = 600 ~ 750 LDC (high)
25% * 2000 ~ 2500 = 500 ~ 625 LDC (high)
10% * 3750 ~ 4700 = 375 ~ 470 LDC (low)
That was interesting. My above table actually suggests that in terms of lifetime usage a medium discharge is most effective in terms of total usage time. The data is based on battery university's findings, lifetime is seen as the time taken to reach a capacity 70% that of the advertised current capacity. Note the 70%, so in tertms of capicity drop offs elsewhere along the scale, the effects could be different. There is balso temperature of storage and charging voltages and currents, a lithium ion charging profile is complex and should usually be left to a store bought charger due to the fire hazard. Keep your battery cool, this is excellent for preserving capacity.
Choosing a voltage cut off that is too high is inconvenient as this will limit time between charges. So you need to know the cutoff voltage at which your amp and speaker cease to work. Please test this with you power supply unit.
Thanks,