Cloud9, you keep mentioning a shut-off function and you know, to be honest, I don't see anything that I would call shut-off. What I see is a charger that applies a relatively constant voltage (subject to Roffs analysis which I suspect will tell us more about how the charger is dependent on load) and just keeps pushing current into a load until the load's voltage fights back and matches the charger's voltage. The fact of life with lead acid batteries is that it is quite OK to trickle charge them all the time and that is what happens with this charger I think. It just brings the battery up to its set-point (say, 13.6 V) and as the battery gets near full charge and reaches this voltage, the charger just keeps it there.
Lead acid batteries, like SLA types, don't suffer from this as long as the trickle charge current is fairly low, which it will be if you set the voltage correctly. Such batteries, while charging, typically will go up to about 14.5 or 14.6 volts if the charger puts out a voltage higher than this. It is not wise to charge them above this voltage. If you then remove the charger, their terminal voltage (even with no load) will drop back down to the natural cell voltage which is much closer to 12.0 volts I think. For example, in a car, if the motor is running (and the alternator is charging the battery) you should read 13.6 volts at the battery terminals. The moment you shut the car off (and the lights and everything else in the car) your battery voltage should drop to about 12 V.
I recommend that you set your charger output voltage at about 14.0 to 14.3 volts open circuit. Then, if you connect a battery thats a bit flat, with say a terminal voltage of 11 volts, then you can expect a current of roughly 3 volts divided by all the circuit and battery internal resistances to flow.
You might also want to read some application notes from a major lead-acid battery maker on how to charge their batteries. One good maker is Enersys who I believe also own Genesis, Optima, and Hawker batteries, all very good batteries.
Getting back to that cut-off function for a moment, it is possible that the little bit of hysterisis that I mentioned before is their way of slightly shifting the end-point as the battery charge current ramps down slowly so that when the charge current drops below 560 mA the effect at VR1 of the fast charge detector changing state is a little bit like somebody adjusted VR1 slightly CCW which would force the endpoint voltage slightly lower and drop the charge suddenly down to a lower trickle rate, lower than 560 mA. So I think this is their simplified version of shut-off, the charge current drops down to a trickle rate and just keeps trickle charging.
You are too focussed on that C3. The time constant that it brings is roughly 100 microseconds, so it doesn't do all that much, just slows down the rate of change of U1B to avoid glitching I think. It is important in maintaining stability of the system I think, but for just testing if the system is working or not, don't worry about it.
Now, I'd like to hear what Roff has learned about the effect of the battery being attached in his simulation.