It is difficult for us to look at the circuit as drawn. The wiring is done as a wiring diagram, rather than a circuit diagram (like mine), so currents run around in circles rather than top to bottom.
Anyway, as a general consideration, an SCR is not a resistive device and will not act as a zero-volt-drop switch (ever). A bipolar transistor is in the same category. The volt drop across it is always non-zero so you won't measure the battery voltage accurately.
A FET on the other hand is resistive so the volt drop can be negligibly small at low currents. I used relay contacts because they are zero-volt-drop and require no power to keep the contacts closed.
So, if I have this right the spec is:
(1) The meter measures the voltage when the ON/OFF switch is on
(2) The meter also measures the charger voltage when the charger is plugged in
(3) The charger needs to be directly connected across the battery (without a diode in the way).
(4) The amplifier must not be allowed to take power when the charger is plugged in.
It would be nice if the DC-DC converter could have an enable control to shut if off, rather than having to put a switch in series with it.
We have to use the charger switch, somehow, since it is the only way to know if the charger is plugged in. Of course this doesn't mean that it is actually charging because somebody could plug in the charger socket but not plug the charger into the mains!
Talking about basics, it seems to me that your circuits do not charge the battery.