Offhand I'd say the weight remains the same. because of conservation. Put a bulb across a battery and the electrons leaving one terminal must equal the number of electors returning through the other terminal. If not you would be creating or destroying energy in the bulb.
There is no law of conservation of mass, there is a law of conservation of energy.
So correct answer is "Yes, charged battery is a tiny bit heavier".
Charged capacitor is heavier because electric field in charged capacitor has energy, hence mass.
With batteries, they they change their chemical composition when charged/discharged, which again corresponds to change in total energy.
LOT is unconditional in electricity and electronics. Current in one terminal must equal current out the other for any form of sources. Any exception would be Nobel material.
Again to absolutely prove it one way or another you have to go through the chemical equations that show energy and mass. Jn thermodynamics it is called a continuity equation, where mass and energy comes from and goes to in any process.
If you want to seriously debate whether LOT applies in the case of a battery then you are into the realm of pseudoscience..
Energy in the bulb shows up as an energy debit in the battery. It would be the same for a simple resistor. The resistor heats up and increases thermal radiation. An incandescent bulbis thermal black body radiation. Electrons are not consumed.
If you measure current on one side of the bulb or a resistor and it is different than the other side that would be worth a paper. The bulb is an ohmic resistor. A piece of wire.