A BMS balances by trimming down the cells with higher voltages to match the cells with lower voltages.
This occurs when charging stops. It can be a slow process since the balance current of most bms is very low.
It does not balance by limiting charge and will only limit charge current as a safety measure if the cell voltages rise above the overcharge detection value of the bms.
The problem of your cell groups not charging evenly is not likely the fault of the bms.
Assuming that your using all new quality cells, the problem could be caused by bad weld connections in the groups that charged fully before the rest of the groups.
Bad weld connections could cause a reduction of capacity and in turn cause faster charging and discharging.
If your weld connections check out, then you could have one or more bad cells causing a reduction of capacity in the group or groups that are charging faster than the rest.
It’s a lot of work when building a Li-ion pack but I would test every cell for internal resistance and capacity before building the pack.
As for your charger going green before the pack is fully charged, check the output voltage of the charger while charging. If it is not 50.4v under load, it will not fully charge the pack.