From my understanding of the problem, you're making it harder on yourself by thinking about a physical manifestation...
Among the abstractions we make in circuit analysis are ideal, independent sources. In the schematic you've posted, the alternator is considered an ideal, 35A current source. This means that whatever it's connected to, it will deliver 35A no matter what. As such, a current source that is not part of a closed loop is a contradiction (because no current can flow through an open circuit).
Similarly, that 12V voltage source is an ideal voltage source, that will always have 12V across its terminals. Similar to a current source without a loop between its terminals, a voltage source with its terminals shorted is a contradiction (as anything connected between two ends of a wire in a schematic are supposed to be at the exact same potential).
With that said, it's not particularly "correct" to think of the electrons coming out of the current source to have a specific potential energy. You lose a lot of the physical world when you abstract a circuit to a simple schematic like posted.
In the schematic posted, and assuming the sources ideal, given that the load has 30A passing through it, then there are 5A passing through the battery (the 12V source and the 100m\$\Omega\$ internal resistance) since KCL must be satisfied. Given that the power delivered/absorbed by a device is defined to be the product of voltage and current, it should be clear that for a given current, different voltages will result in different powers. In this case, the total battery voltage is 12V + (100m\$\Omega\$)(5A) = 12.5V. Thus the battery is, in this case, absorbing positive power with a value of (12.5V)\$*\$(5A) = 62.5W.
With a 6V battery, assuming the same conditions (meaning the load is still pulling 30A and the internal resistance of the battery is identical to the 12V battery--which means the load must be different, by the way), you can see how the power absorbed by the battery would be smaller.
For further thinking... consider a simple loop consisting of a voltage source and a current source, any value for either (for instance, say 1V and 1A, respectively), and both of them connected following the passive sign convention (current source pointing from (-) to (+) on voltage source). By the definitions put into place in circuit analysis, there must be 1A flowing through the loop, so the 1V source is absorbing 1W of power, while the 1A source is delivering 1A of power. Change the 1V source to, say, a 5kV source, and the power absorbed/delivered will be greater by a factor of 5000. Note that the electrons coming out of the current source now must have 5000 times the potential (by definition, since the voltage source has 5000 times the potential across it), but the current source hasn't changed. Naturally, if the 1A current source were not ideal, it may not be able to supply 1A through a 5kV battery, but in circuit analysis, these ideal sources are defined to deliver their rated spec, regardless of the circuit they are a part of (short of contradictions).
I can't tell you what the problem is with your apparatus since the photos can only give a limited amount of information.
However, it appears from the photo that you just have a voltage source (battery) in series with a resistor in series with an ammeter (you do have the multimeter set up to measure current and the leads connected to the proper inputs, correct?).
Now, this is certainly a valid connection and, if the above is a correct description, you should be able to measure different current values for different resistor values (however, if your multimeter and leads are set to measure voltage instead, you will only measure the battery voltage regardless of the resistor value).
However, with such a simple circuit, you can simply use Ohm's law to calculate the current.
Since an ammeter is effectively a short circuit, the current through the resistor is simply the battery voltage divided by the resistance.
In other words, if you simply connect the resistor directly across the battery and then measure the voltage across the parallel combination, the current is simply the measured voltage divided by the measured resistance.
Best Answer
There's the usual, droll electronic laws answer. But I'll take a different perspective that I think gets the point across more deeply.
Take a capacitor. You know that it can store charge, right?
So, given your thinking process, you might then imagine that the current on one of the terminals can be different from the current at the other terminal of the capacitor. And you might ask, why must it always be that the current goes through the capacitor instead of having current X leaving the left terminal of the capacitor and current Y entering the right terminal of the capacitor, where \$X \ne Y\$. Right? I mean, you would think this could possibly happen?
Not usually. Just possibly?
Or, perhaps, that we might set up two capacitors in series (like your batteries) and where you would imagine that one of them could have a different current in it than the other one. One could "build up charge" so to speak, you might argue.
Yes?
No.
Sometimes, it is very hard for people to realize this. But the force acting between two charges is huge. Not just big. Not humongous. But unimaginably huge. The effect is that matter in nature is essentially neutral. At all times. Ionization does occur. But it is ever only for very small numbers of charges.
Let's take your two batteries. They are separated, let's say, by a distance of \$10\:\text{cm}\$. They are each \$10\:\text{cm}\$ wide, too. So the mean distance between their centers is \$20\:\text{cm}\$. Let's say that the current leaving and entering one of these batteries is \$1\:\text{A}\$ and the current leaving and entering the other one (in series, as you say) is \$2\:\text{A}\$. Then it must be the case that a charge differential is building up between them, based up the missing \$1\:\text{A}\$ that is accumulating into one of these batteries.
(Keep in mind we are assuming magic batteries here that can actually manage to keep and hold a lot of charge.)
How long might you wait until the force acting between these two batteries is 2000 lbs, or one ton!! Well, it's \$20\:\text{cm}\cdot\sqrt{\frac{8900\:\text{N}}{k_e}}\approx 200\:\mu\text{C}\$. At the differential rate of \$1\:\text{A}\$, this would take about \$200\:\mu\text{s}\$. In that short time, assuming magic batteries here, you'd already have an incredible force acting between those batteries.
In reality, that of course does not happen.
There can be moments when things are not exactly in balance. But these are for very short moments while charges redistribute.
The universe needs pretty much everything in it to be relatively neutral.
Gravity on the other hand is pathetically weak. You can accumulate a lot of mass before any serious force occurs. But with electric charges?? WoW! No way. Everything stays pretty much neutral pretty much all the time.
At least, on Earth.
So why do all the currents summing into a node have to equal all the currents exiting the same node?? Because if that didn't happen, it would take no time at all for the circuit to be literally attracting nearby planets towards us.. or perhaps repelling the Earth from the sun at a fast pace.
It would be cool, I suppose. That kind of power would be nifty to possess. Rocket ships would be trivial to launch. Just turn on the current for a little bit and whoosh!! Up into space they'd go.
Life would be so much different.
But then we'd probably not be here, either.