Are charged batteries heavier?

[mfb]

What if instead of weighing batteries, you could prove one was lighter and by roughly how much...

Measuring masses and mass differences is commonly known as weighing...

To avoid material changes, you could discharge the battery via a resistor. Measure the full battery plus resistor, wait until it is discharged and cooled down and measure again.

 

[Derek P]

So, to bring this back to the OP's question:
the inflow of electricity, in the form of electrons, causes molecules to change shape, and some of the electrons' orbital energy is converted back to mass ... by way of muons?

Or am I word salading here?

Word salad with mayonnaise :) Muons and electrons are different fundamental particles. Unless your battery has a built-in particle accelerator, it's not going to change the particles themselves. Electrons remain electrons.

The energy of interest is indeed the orbital energy but it is not simply kinetic, it includes (and arguably it is) the energy of the electromagnetic field that holds the molecules together. And a field that has energy must have mass. E=mc^2.