Hi.
Do you know a reference that addresses this specific point?
Almost exactly this question has been discussed on these websites (all forums or Question and answer sites):
https://physics.stackexchange.com/questions/34421/does-the-mass-of-a-battery-change-when-charged-discharged
https://www.quora.com/Is-a-charged-battery-heavier-than-a-depleted-one
https://www.physicsforums.com/threads/are-charged-batteries-heavier.940974/
Some of these have additional links to other information sources. The theory is so widely accepted that you can sensibly take for granted that it is true - a charged mobile phone battery would have more mass than an uncharged one - but there is very little direct experimental measurement: Most people agree that for a typical household battery like a mobile phone battery, there isn't any laboratory experiment that would reliably detect the change in mass - because the difference would be so small. That may change in the future but at the moment that does seem to be how things are. We do have experimental evidence for systems that absorb or release larger amounts of energy and then show a mass change - the obvious example being nuclear reactions in power stations and weapons.
The idea of a charged battery having more mass is sufficiently accepted that it is even used as an example in some texts discussing mass-energy equivalence:
The value of the mass of the hot brick or charged battery is greater than it was before it absorbed energy.
[Taken from: https://plato.stanford.edu/entries/equivME/ ]
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A different way of thinking: Stop worrying if some energy will show up as mass. Instead realise that is extremely rare for there to be any mass that isn't explained as a form of energy.
"Mass" just isn't what we thought it was about 50 years ago. For example, the Higgs field provides a description for something that we might reasonably consider as the intrinsic mass of a particle but even this does not help very much. The mass of most particles is far greater than the mass explained by the Higgs field. For example, a proton has only about 1% of its mass explained by the Higgs field the remaining 99% is the consequence of other forms of energy being located or contained in the "system" that we describe as being the particle. Most of a protons mass isn't the Higgs field mass of the 3 quarks in it, instead it's due to the binding energy that is present when the 3 quarks are in such close proximity with each other.
To paraphrase this, at a microscopic level, it's actually better to think that all "mass" is just "energy" of one form or another bound up in one microscopic system that we call a particle.
Here's a short video by a slightly flamboyant presenter (under 7 minutes) that discusses the issue:
"What the heck is mass?" - Science Asylum, available on You Tube.
Here's a longer video (if you have 40 minutes and a deeper interest):
"The Concept of mass" - presented by Jim Bagshot for the Royal Institution. Available on youtube.
Best Wishes.