Not quite; you show a much higher charge voltage and several diodes.
Basically, the diodes aren't needed, the charger needs to handle the full load (4A, not a 2A step-up), and the charger needs to be a lower nominal voltage (so as to not overcook the battery).
The LV cutoff can be present regardless -- before or after the final diode(s) -- because as long as the charger is around, duh, sufficient output voltage is available.
It... makes me think about ways of thinking:
One way of approaching a problem, is to list all of the possible states, and assume they are exclusive; then the solution is a state machine which takes inputs, and transitions between those states conditionally. This is a common trope just in ordinary life I think (do this, do that, etc.; do/don't do that other thingy if you have the stuff to do it, etc...). It's also specific to software (procedural). But it misses facts about the underlying systems, or components, or requirements.
So there's another approach, which is to cut through the state machine, and see what the lowest, basest, simplest realization is, and understand it in terms of the components used to build it.
We think of batteries as charging, idling, discharging, etc.; so we are tempted to think about implementing all those as discrete states, and so we toss in so many diodes and switches and controllers to do that.
But the battery really doesn't care, it's just a pile of gloop with wires stuck into it.
Charging and discharging are just two regions of a continuum variable, the terminal current.
What if we normally power the load from a power supply, and just happen to shove a battery in there? Well, we do want to check that the power supply doesn't draw any power from the battery -- which is usually the case (the PSU's output stage is typically just diodes). We also need to check that the supply can deliver full load current, so we aren't accidentally discharging the battery even while we have full power available. (Or perhaps we do, because we want to supply peak loads at higher capacity than we have available otherwise?) Finally, we need to check that the battery's rated charge and discharge cycles are compatible with this kind of treatment. Which for lead acid, is absolutely fine.
We might consider a more complicated circuit for Li ion, where the charge current should be limited (and, I think they're not recommended for float duty?, but it can be done); or for NiMH, where the charge voltage is indeterminate (they cannot be floated, they'll take no charge, or overheat and dry out).
For that case, we should use a self-cycling charger, and that's it for the battery (well, and undervoltage or overcurrent protection on the load side). Then we effectively wire the battery in parallel with the main power supply, using a wired-OR controller (or diodes if we don't mind the voltage drop). For this to work, the power supply must be higher than the battery voltage. The charger can be powered by the main power supply; simply check that there's no condition where the load is powered through the battery and charger (probably slowing or wasting charge, and costing efficiency from the extra power stage).
For a power supply comparable to or lower than the battery voltage, a wired-OR controller won't work, and active switchover is necessary. The load should be tolerant of momentary dropouts (might be microseconds to milliseconds), while the switch operates. Presumably the switch will be triggered by lack of POWERGOOD from the supply. This has a negative resistance characteristic (when the switchover occurs, power supply output voltage momentarily dips as it adjusts to the new load; which can trigger POWERGOOD dropping momentarily, which triggers..), so the whole loop must be checked out carefully to ensure stable operation. Delays might be built in to account for those response times, and a possible consequence is simply that the load may need to tolerate dropouts.
Basically, this whole mess of a process is why laptops typically use 18-20V supplies, and 14V battery packs. The changeover is seamless and passive, you can just unplug and walk off with it.
Tim