Battery balancing circuit

You are much better off making ideal diodes using MOSFETs

• Replaces a Power Schottky Diode
• Internal 20mΩ N-Channel MOSFET
• 0.5μs Turn-Off Time Limits Peak Fault Current
• Operating Voltage Range: 9V to 26.5V
• Smooth Switchover without Oscillation
• No Reverse DC Current

But the simplest solution is a better Schottky Diode Array $6 with a heatsink

These are common cathode.

Charging one battery from the other will result in overall drop in efficiency and run time. Consider that energy will be lost in the charging circuit and in heat losses of both the battery powering the charger and the battery being charged.

The most efficient is to take power from the 'good' battery. As its voltage drops to the level of the battery with lower voltage they will both support the load.

When both are charged up your diode arrangement will work and is the simplest solution. With this arrangement the battery voltages will drop at the same rate.

You say series is out... but may have dismissed that too quickly.

You have not really detailed what the power stage requires but I am assuming in this answer it expects some battery like voltage. You have also intimated that the double-up is to extend duration not increase the current available.

As such you may want to consider putting the batteries in series and adding a buck-boost invertor to regulate the output to your required voltage. These can be quite high efficiency and will supply you with your required voltage even when the batteries have fallen significantly below the required output voltage. The power gained from the latter may be enough to compensate for the losses in the regulator.