My thought was that LT3081 is a floating regulator just like TL431 so no ground pin & therefore could be used connected in series with each LT3081 charging its equivalent battery in the series - just like with TL431.
The TL431s are being used as power shunt regulators so they only see the voltage across them. Essentially they are just high power precision zener diodes. You might be able to wire an LT3081 to be a power shunt regulator but the result will not be what you want.
AFAIK, there's no need for multiple floating supplies in this scenario.
No, there isn't, but it is the easiest way.
If LT3081s are used then each has a ground (really common) reference which is either the ground (common) of all of the cells or just the one cell that it is attached to. But the input voltage for each LT3081 has to come from somewhere and if there is only one power supply, then it will be referenced to the ground (common) of all of the cells. The power supply might have multiple outputs to supply each LT3081 so each operates with a minimum of power dissipation but there is another problem.
The charging current through the top cells flows through the bottom cells. This would be acceptable if there was always a load on the bottom cells but without a load, cell imbalance is likely unlike with the case of multiple TL431 shunt regulators individually protecting each cell. The LT3081 has no ability to sink current like a TL431 or any shunt regulator.
The multiple floating power supply suggestion solves that by referencing each floating supply output to only the individual cell that it is charging.
The issue with using one LT3081 across the series of batteries is that I may want to pull voltage output @ ~1A from any of the middle batteries in the series (i.e a different voltage output) & I haven't seen a balance circuit capable of keeping the batteries in balance with that sort of current draw. Maybe I'm wrong?
I have not seen a balance circuit capable of high currents either but that is just because there has been no need for it. An LT3081 is half of a linear balancing circuit; it just can only balance in one way. A micropower operational amplifier with a pair of power transistors can easily drive an amp or more of balancing current between two cells while drawing nothing when no balancing is required. The transistors (1) only have one cell worth of voltage across them so at 1 amp, power a little less than 4 watts which is not difficult to handle.
Or keep the TL431 power shunt regulator string for its simplicity because it will do the same thing. The operational amplifier way might be perfered for lower quiescent current draw.
(1) This might be a good application for complementary NPN/PNP Darlington pairs.