Does your circuit behave the same if you remove LTC4412? How much current is drawn by Battery when it is being charged by 12VDC?
The charge current is 600mA and the maximum current drawn by the internal circuitry is 1A.
With the LTC4412 is removed, the circuit is working fine. When 12VDC and battery, both are present,the 12VDC is feeding the internal circuitry. In absence of the 12VDC, the battery is feeding the internal circuit. I think the battery is feeding the internal circuit through the body diode of the MOSFET, FDN306P (Q2).
You may want to check VIN/2 waveform using oscilloscope to see what's happening there when 12VDC is present.