Few problems with your understanding, and hence implementation.
1) Cells are nominal 3.7V, min 3.0V, max/charge 4.2V, so 3S is 11.1V, 9V, 12.6V.
So charging a heavily depleted battery from 12-14V input needs buck, charging a battery with higher state of charge need boost.
The charger does buck internally, this was more of a given since a battery charger by design has to step down voltage.
My comment was that the charger needs to boost the 12V coming in so a higher voltage so it can charge a 3S lipo above 12V input. This sort of lipo charger with boost is common in the RC world where you can charge a 6S lipo from 12V DC wall adapter.
2) What sort of voltage range is your home automation control system rated to ?
Currently the 12V bus is used to run 12V door locks and some step down regs for other things. Those are rated for 15V.
The door locks are probably ok up to 13.8V
3) no figure is given for the size of the cells or the amount of current you system draws.
Doesn't really matter, i gave the charge current needed as 0.5 - 2A.
System draw is complex as it depends what its doing and it will shutdown the SBC's if mains fails. In that state it just needs 500mA for door locks and 100mA for primary control MCU
4) There is no "float charge" for Lithium Ion, unless you want to damage the cells. They are pretty much Constant Voltage charging with current limit, the stop charging point is when current drops to a low value (75mA is a commonly given value).
You can float charge a lipo easily as long as you don't need full capacity. You are simply charging them up to a set voltage which is under their 4.2V max. 3.9V for example which provides long lifespan.
The current goes to zero once they hit 3.9V and charging is stopped by design.
An easy way out is to get a LiFePO4 SLA replacement battery, they range from not cheap to silly prices. However it's a black box, with self contained electronics to manage/protect the cell. Also they are configured as 4S 3.2V nominal per cell.
That is useful to know, i didnt know they existed. Thanks