A commercial 'battery tender' device I'm using has two lights: red for 'charging' and green for 'floating'. It apparently has a one-second apply-full-charging-current test, and on exceeding the voltage threshold, turns current (and the red LED) off, then looks at terminal voltage... if there's no battery, or a low voltage, it then reapplies full charge current. So, on battery-open failure, it blinks a red light. If battery is low, it trickle-charges (red light) at limited current until it hits the high charge voltage threshold, then the red light goes off, and green LED goes on, as it falls back to float at some limited voltage.
A little clocked logic, a voltage reference, and a couple of switch transistors, with a comparator or three, can easily implement these functions, when given a suitable current-limited power source. A single comparator and a bandgap reference, with some shunts on tri-stated outputs, makes a multithreshold voltage tester.
For a solar power source, it would be useful to add a crude PWM boost so that the sub-12V hours of daylight availability don't get wasted. The most effective way to handle that, might be a microprocessor. With a PWM boost, you could use lower-voltage solar cells and do all the current limiting in software.