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.
It is very common to connect lead-acid batteries in parallel to increase capacity.
A common house battery bank on pleasure boats consists of four or more, 6 volt "golf cart" batteries connected in series/parallel to make a 12 volt battery bank. There is no need for diodes or other isolation between batteries.
Some people express concern that a shorted cell in one battery will cause overheating of the wireing, but I have had a couple of cases of a shorted cell with no problem other than being unable to fully charge the bank.
I have no experience with LiFe batteries.
Best Answer
No, a diode circuit as you describe won't automatically balance the load across the batteries in the manner you describe.
Note that everything below this comment only applies if those packs are raw LiIon packs with no internal regulation. If they're internally regulated, then all bets are off.
It'll work more to keep the state of charge of all four batteries more or less equal as they discharge. However, if you have three bad batteries and one good one, that good one will take the load. If you have three halfway discharged batteries and one fully charged one, that fully charged one will take the load. Etc.
There's a good chance that if you take pains to get lots of four batteries and use them together (i.e., charge them at the same time and always load all four at once into your rig) that a simple circuit as you describe will mostly do what you want, because the batteries' voltages and the diodes' drops will be close enough together that the batteries' internal resistances will handle the match.
If you want to absolutely enforce a rule to "don't take more than 10A from any battery, ever", then you'll need some sort of fancy power supply management circuit, and it'll probably have to be custom made for your purpose.