Some other possibilities:
With the power supply un-powered on the AC side, connect it to a charged lead-acid battery (should have an open-circuit voltage of 12.65V at room temperature). Measure how much current leaks backwards from the battery into the power supply. If it is under 1mA, you do not need a diode between the supply and the loads.
If the power supply voltage is well-regulated, and at no-load its open-circuit voltage is less than 13.20V, you do not need a diode between the battery and the loads. If the supply is adjustable, diddle it down to 13.10V, leave out the diode, and the power supply can be left on 24/7 to act as a float charger for the battery. At this voltage setting, the power supply will not charge the battery, but it will act to keep the battery from self-discharging over the winter months. As a minimum, diddle the supply voltage to overcome the forward voltage drop of the diode if you need it.
I used to have a RV where I used a large 35A switcher to run the trailer loads from AC, recharge the battery, and float-charge the battery during winter. I built a controller that automatically sensed the battery voltage and current, and it would run the battery voltage up to 14.5V while the battery was accepting charge, then it would lower the voltage to 13.1V to float the battery after the battery was fully charged. My add-on controller effectively implemented the three-step lead-acid charging algorithm. You can buy such RV battery charger/power read made. No diodes required.
Another possibility: Add a high-current, AC coil SPDT relay. Use the presence of the AC to pull-in the relay. Use the SPDT contacts to select between battery and power supply. No power loss between the battery and the loads that way...