Personally, I'm having a hard time accepting it for several reasons.
First of all, there's the issue of the system. Sure, it's all in good taste to say that a battery is a few fractions of a gram heavier when full - but let's face it, it's pretty much impossible for a battery to lose charge outside of a system (a circuit, or charge degradation by chemical or heat loss). As such, I am 90% certain that there is absolutely no energy loss within a closed system (most circuits are technically open, energy-wise... and even then, technically speaking, the only direct way to lose energy is via the electromagnetic field - induction or light. Because out of an atmosphere, heat is retained, and otherwise very little energy is lost).
For instance, with this car, the only losses there was were technically thermal: by accelerating, they converted electric energy to thermal and kinetic. By braking, they turned that kinetic energy into more heat (and some electricity, as IIRC the Tesla Model S has regenerative braking).
And hell - at a very basic level, let's talk about charge degradation. There's two things that happen, be it ion-based or chemical-based, and one is the result of the other. The main thing that happens, when a battery loses charge, is quite simply what I call 'internal resolution' - the charges balance out or the chemical reacts to become a neutral substance. Sure, it loses charge - but the laws of thermodynamics say that you can't just lose energy. So where does that energy go? Two ways: a major portion of it is transformed from electric energy to chemical energy - almost all of the battery's power could, most likely, be found within the chemical bonds of the workings themselves (or, in case of ion batteries, the sum of the electric charge be found in the electron layers of the atoms).
Second issue I see with the statement is that, mathematically speaking, the mass loss is insignificant - and so the only two reasons why we should even CONSIDER that being a valid event is if dealing with power sources of absurdly high energy density (quick math gives me 90 petajoules of stored energy to gain a kilogram - so assuming a significance of 0.01% and assuming that my math is right, for energy usage to be significant we'd need to be dealing with something akin to 9 terajoules per kilogram of specific energy - which means we would be having a battery with a tenth of the energy density of an uranium breeder reactor, twice the density of a tritium reactor - and that would be about two order of magnitudes better than what non-rechargable lithium batteries are capable of today).
Third issue is applying relativistic mathematics to events and objects that are not even close to having relativistic vectors. It's like making a statement that the Earth's orbit was altered by one centimeter at any point - sure, it MIGHT have happened, but it serves no purpose and given the units of measurements is completely irrelevant.
So sure. It's good food for thought, but completely irrelevant to all modern day applications (and might always be).