Can I safely balance a series/parallel cell circuit to power a 12V soldering iron?
Samuel, you seemed to be treating a pure DC circuit with a resistive load like it was an AC circuit with a reactive (eg. inductive) load, and like there was an amplifier in there somewhere with an &quot;inverting&quot; input.  That noted, the drawing currently up is a straightforward matter, so I think you are on a better track now.
I would just put three 4.2 volt batteries in series and forget the balancing act you are trying to do.  The rheostat won't help unless you are using it to adjust the soldering iron power (but that's a terrible way to do it).
Careful calculations cannot preclude variability between batteries, which is usually substantial, and which is the reason that one cell in a series battery pack can be damaged.  The cell with the lowest capacity dies first then the other cells push current through it, which pushes its polarity into reversal, destroying its chemistry (and possibly even making it leak or explode or burn).  The best simple protection against this happening is to be sure that all the batteries are as evenly matched as possible, and never discharging the pack very far.  I see by your reasoning that you understand this, but possibly do not understand that battery capacity ratings are only approximations.  You cannot guarantee much by trying to balance capacity ratings.  Balancing is more for the sake of making the whole thing work as long possible, rather than only working until the weakest battery dies.  Without a more sophisticated battery management system -- a much more complex circuit -- you are going to be stuck with simply being sure you recharge all the cells before any of them die.
This is made more critical with lithium batteries (compared to lead acid or alkaline) because they have very flat discharge curves.  The one cell with the least capacity will drop voltage rapidly when it meets the end of its capacity while the others in the battery will still be pumping out electricity at almost full power.  So you have one flat battery that is easy to be pushed to death by the remaining full-power batteries.
For these reasons I have to say &quot;no, it is not safe to do that with lithium batteries&quot;.  If you ignore that warning and go ahead anyway, carefully watch all three cells to make sure you stop using the battery if one of them is exhausted before the others.  Since lithium cells have a cutoff voltage around 2.5 volts, you probably want to stop using the pack well before any of them go that low while under load.  Definitely don't wait for one of them to go even lower.