The AC source won't need another diode since it goes through a diode rectifier.
I don't recommend a Schottkey. They can have significant reverse bias leakage current that will go into the battery which may harm a non-rechargeable battery over several months.
Watch how the filter cap works carefully. If there is so much ripple that the cap voltage drops below (battery voltage - 0.7v) at any time, then it will discharge the battery for a moment during every cycle.
Simply connect an ammeter in series with your battery, measure the current on mains (should be zero), and the current on battery - which should be more than zero.
Ammeter, isn't that the tool you use to find out how much current there is left in a battery? And shoulnd't the bettery read zero while the main should read anything above zero? Other way around of what you wrote.
ammeter tells you how much current is flowing, not how much is left. If you have the device plugged into the wall, battery current should be zero and mains current should be non-zero. If unplugged, mains should be 0 and battery current be non-zero