Parallel power sources with battery priority

Main project requirements and background

Because "free" energy is available from batteries charged by a solar panel, an automatic power-source switching system is required: the load should be powered from the battery first and, once the battery is discharged, from the AC-DC power adapter.
In my case, the load is 12V LED lighting.

The circuit is powered on by switching on the AC-DC adapter, which turns on MOSFET Q1.
An 8.13V Zener diode reduces the voltage applied to the MOSFET gate. Together with resistor R2 (150R), it helps discharge the capacitors in the AC-DC adapter and turn the MOSFET off faster when power is removed. This is necessary when the LEDs are powered by the battery.
N-channel MOSFET Q1 (IRF3205) is needed to create a closed circuit from the battery through the relay coil and the MOSFET.
If the battery is discharged or absent, the DPDT relay is not energized, and current flows through the normally closed contacts by default, powering the LED from the 12V AC-DC adapter.
If the battery is connected and charged, energizing the relay switches its contacts, and the load (LED) is powered by the battery.
The DPDT relay coil operates from 9.5V (rated at 12V). I used a relay with two pairs of contacts. For this project, the second pair is not used.
Diode 1N4007 protects the MOSFET when power to the relay coil is interrupted.
Schottky diode 1N5820 prevents the relay from repeatedly cycling on and off.

A list of components used in this project, with links, is available in the Google Sheet ("Preferential battery supply" tab).