I've seen similar trickle chargers. Since the output is unregulated you shouldn't call it a maintainer. If you leave it connected for an extended period it can kill the battery. What model is it?
The fuse is actually a self-reseting circuit breaker. If the battery draws too much current, the breaker heats up and pops. When it cools down it resets. If the battery is deeply discharged, this is likely to happen over and over. Eventually, after a number of years, the breaker just dies. The current value should be shown on the unit. It should be similar to the 3A rating of the 1N5404 diodes. If there's no continuity through the breaker at room temperature, you have to replace it. A good auto parts store should have them in whatever value you need.
The voltage output is misleading. Since the transformer is putting out 13.8V AC, the peak voltage is actually 19.5V. Subtract a volt or so for the diode drop and you're looking at an unregulated output of about 18.5V. Normally, no capacitors are needed or used, but you can confirm the peak output voltage by putting an electrolytic capacitor of almost any value across the output. Watch the voltage rating and polarity of the capacitor! You should see a voltage of about 18.5V DC. This is why you can't leave it connected for a long time. 12V batteries aren't impressed by charging voltages of 18.5V!
Since the transformer is putting out voltage and the diodes are good, have you tested the breaker?
Ed