Something of a grab bag here to get things started.
Generally, the jellybean solutions are for single cells only. And certainly a multiple-battery swap system is beyond the usual scope of generic designs.
Combining the battery outputs with Schottky diodes is a classic move. You can have a look for Ideal Diode or ORing or Hot Swap controller chips from several companies to do a better job (e.g. less voltage drop, more control).
Your circuit doesn’t seem to have any over / undervoltage protection on the cells. This is important for safety!
I see you have a ‘12V’ input. If we assume a 10% drop for tolerances and cable resistance, you only have 10.8V, which isn’t enough for 3x LiIon cells (might be OK for 3x LiFePO4).
FYI, if you’d like to ship a product with Li batteries in, there are limits to the size of the cells and total weight of cells.
Further, if you want to take the product on an aircraft in your hand luggage, the maximum allowable battery pack size is 100 Wh.
The way you tap the system load off the chargers can be a little problematic, because it means your chargers don’t have a measurement of the true battery current. What are your charging and system load currents?