What colour is the magic smoke when it vents with flame ? :-)
Rushing immensely, more later, but ...
What you describe seems to run a severe risk of doing damage.
To be ure, first you need to specify the allowed MAX charge rate for each LiPo.
You have 1.0 + 1.2 = 2.2 Ah in parallel so 5A = 5/2.2 = 2.27C.
This MAY be OK if cells are specd as 2C or more and are balanced in draw.
If specd at say 10C then it all may survive.
If specd at 1C it is very very bad.
BUT when a cell pair plateaus at 8.4V it's current will start to drop and if the charger is able to make 5A the extra current WILL flow into the still in current more other battery pair.
If max charge rates are not >> 2C then what you describe is at best an extremely poor compromise and at worst a disaster either in magic smoking or in cell lifetimes.
If max charge rates are around 2C then what you describe is at best a beating of some of the cells regularly and at worst a journey towards magic smokedom.
In an arrangement like this with different capacity pairs wired in parallel you need to carefully monitor individual cell pair or even cell voltages to prevent discharge-damage. This is going to make balancing more important, although I have been impressed with how well cells from the same batch seem to track when I have checked it (not often).
Operating cells of different capacities in parallel is an immednsely bad idea usually unless you manage and turn off each pair individually.
Apart from one pair endpointing before the other and throwing more charge or discharge onto the other there is a lack of certainty re how cells load share.
eg Say you have a 1000 and 1200 mAh cell and load both with 1000 mA. The large cell will see this as less of a percentage load so it's natural terminal voltage will be larger and it will "happily" supply the extra current. but there is no guarantee that it will o this in the ratio of the tywo capacities. The large cell may prove very "sacrificial and provide most of the load for most of its capacity. BUT when it finally falters the small cell will then take up most of the load and may now be overloaded. And there is no certainty that the LARGE cells will not now expire and be driven into a damaging mode. Probably not, but. too many uncertainties.
Why run cells of different sizes and in this 2 x 2 pattern?
Key question: What are the max allowable charge current rates for the 1000 and 1200 mAh cells.
Without this information the question become svery hard to give a good answer to.
This is quite the multipart and deep question. You seem to understand the basics but just in case, I’ll recommend this website as a good overview, albeit dated in terms of current ICs and BMS competitors.
This is dependent on application. For small packs like the one you’ve drawn, there are a wide variety of chips available. For larger packs, Linear (LTC6803) and Maxim (MAX11081) are the two main competing suppliers of BMS solutions. They provide the most integrated solutions for multi-cell management. The main differences from smaller packs are the higher series voltages, daisy chain communication and noise immune signaling.
In either case, voltage monitoring, temperature monitoring and active balancing tend to be the primary needs of a BMS. Other features such as redistribution tend to be less critical and often do not warrant the added cost and hardware.
Unfortunately, even though nearly all BMS designs have the same goals, rarely is there an all in one solution. Solutions are based on number of cells, voltages (chemistry), tolerance for loss, communication method etc. These variables are not all standardized between one project and another. Furthermore, technologies keep changing. Linear is considering replacing CAN-BUS with shielded Ethernet. ADC accuracy is increasing. Sample rates are increasing.
The tried and true method is just simple active balancing. That is extra voltage on a cell is bled off resistively. Other than that, the rest of the system is a simple matter of reading all the cell voltages efficiently.
Eventually, we will see a more robust landscape with better standardization. Companies like Elithion or Nuvation are designing BMS solutions that are nearly drop in solutions. Chip designers are getting a better understanding of customers needs and have already sought to distinguish between the different types of li-ion needs based on different industries. This will mean that we’ll start only paying for features we need. Even battery cell manufacturers are standardizing cell types. For now though, any battery design remains quite customized.
You are much better off making ideal diodes using MOSFETs
But the simplest solution is a better Schottky Diode Array $6 with a heatsink
These are common cathode.