The vendor is right. You must start with cells that are closer together in voltage than what you have now. Your example shows 2.63V to 2.381. As you've discovered, that's never going to work.
I hope this analogy translates OK, but think of the 12 cells as an hourglass timer. That's a piece of glass with a two storage chambers for sand and a funnel in the middle. Start with one chamber full of sand, turn it over and 3 minutes later (or whatever the timer was designed for), all the sand has fallen from the top chamber into the bottom one.
In your bank, you have 12 hourglasses, all must start with the same amount of sand in each chamber. So when discharging, they all run out of sand together, and when you charge, that sand is put back on the opposite side. At all times, the sand on either side should stay the same relative position across all 12 hourglasses. Otherwise, the BMS will stop all the hourglasses when any one of the 12 runs out, either charging, or discharging.
We call this cell balance. It's essential for all lithium banks. The sand is measured by the voltage. You should ideally have every cell read the exact same voltage, but that's near impossible. But if you see the middle cells at 2.2V, the rest should be say 2.18 to 2.22V. That's good enough. Expect more variation at high state and low state of charge.
Fixing your situation isn't easy. What Chalo suggested, a balancer, is one option. It will take some time, and is probably the easiest option. However, if you go this route, I suggest removing it (or turning it off) once balance is restored. Otherwise, you won't know if it's making a bad cell or cells. Once in balance, the bank should stay that way for some time, at least a month, normally at least 6 months with fresh cells. Or forever if the BMS built in balancer works well. If the bank won't stay in balance, more troubleshooting is required.
There are two other methods.
1) take the bank apart, wire all cells in parallel together making one really big 2.2V battery. Then charge to the manufacture's spec full charge voltage, somewhere around 2.6V per the BMS settings. Do the charge slowly, say 1-2A. Leave the cells connected after charging for 8-24 hours. Finally break it back apart and reassemble the 24V bank. This is normally done prior to installation to begin with. Sometimes in groups of cells, but following the exact same charging profile and wait times. This will require a lab type variable power supply, or a well regulated charger intended for this purpose for LTO cells. A lot of guys use chargers from the scale model remote controlled hobby industry.
2) For smaller imbalances, if you have lots of time...you can take 50ft of AWG 18 wire and short out the "high cells" one at a time to match your "lowest cell". Don't leave the wire coiled. This should be a safe rate, but make sure the wire doesn't overheat, warm is expected. More wire length lowers the heat if necessary. It will probably require a couple passes. You must physically monitor this exercise. The BMS won't protect the individual cell being discharged. However if the bank is basically working, this can be done in place, over time. Getting your 2.6V and 2.5V cell down will let the BMS get loads more charge into the bank. I would only suggest this if you have lots of patience. It's a cheap trick for fixing small imbalances, but yours is large. The only advantage is you don't have to break apart the bank or buy and trust a cheap balance board.
Good Luck!
