Yes, it may be possible to charge the series capacitor that you show and then reverse it's connections to place it is series with the battery to get a higher Voltage across the load FOR A SHORT TIME. But here's what the load will see as these processes happen, using YOUR second drawing, which I would modify.
1. First, starting with the capacitor discharged, it will look like a DC short so the load will see the full battery Voltage or +10 V.
2. As the capacitor charges that charge will show as an increasing Voltage that is bucking the battery so the load will see V2 as decreasing from +10 V down to zero or some lower number depending on how long the capacitor charging is allowed to continue.
3. Then the capacitor is reversed so it's +10 V is in series with the battery and the load sees +20 V for an instant.
4. Then the capacitor starts to discharge through the load and battery so it's Voltage decreases and the load sees a Voltage that decreases from the +20 V value back down to the +10 V value.
5. And it repeats. V2 looks something like this:
{My attachment should be HERE! But this stupid BB software does not seem to want that. I give up. It is below}
I would alter your drawing by having the battery Voltage connected to the load while the capacitor is charging. That way the half of my drawing below 10 V would become a straight line at 10 V.
But even that is NOT how I would double the Voltage to the load. I do not see any reason why a floating ground is needed in this circuit. I would use the battery Voltage to run a DC to AC converter which produces an AC Voltage. Then I would use a conventional Voltage doubler circuit (diodes and capacitors) to boost the Voltage to the 20 VDC level. This would include A GROUND connection to the battery minus terminal.
Or, BETTER YET, just use a DC to DC converter in boost mode WITH A GROUND connection to the battery minus terminal.
The "floating ground" buys you nothing in a battery powered circuit. The whole thing can float. It can float on top of 100,000 VDC if you wish and it will still work. Just turn the 100,000 VDC OFF before changing the battery.
By transforming a quadripole into a dipole, it will be hard to get anything else than |V2| <= |V1|.
Maybe we can try leveraging some quantum effect.
So you are saying that there is no way to get V2 above V1 even for a short period of time ?
No, for a transient, the question is obvious and answered already by magic. Just put anything in series which has a charge. It will just be extra energy that has been stored previously though.
But once this charge is depleted, game over. So what does that bring to the table?