Even if they needed that much energy, the best way to do that [that we know of] is to build a Dyson sphere. With every star shrouded by a Dyson sphere you no longer need to shroud the galaxy.
A more critical assumption is that a type III civilization even cares about masking its presence. If they were even worried about being attacked, the speed of light would mask their progress for a great deal of time and, because of the distances involved, even seeing them once the light reaches you would be a massive accomplishment - most likely you'd have to search for individual stars in another galaxy which are disappearing in unusual ways.
A type II civilization could possibly build a telescope with a large enough aperture but I assume that type II civilizations would be long past their ignorant warring ways.
Also, a civilization could have dimantled Andromida Galaxy 2 million years ago and can't tell yet. For more distant objects the gap gets rather wide.
It's entirely possible to burn deuterium in small reactors but there's only about 1/2500 as much deuterium as hydrogen.
(Perhaps some unthinkably advanced civilization could figure out how to get energy from hydrogen fusion in a small reactor, but you did specify "our current understanding of physics.")
As to size, on galactic scales all our current fusion designs are tiny. Scale it up to even just 20 miles on a side and proton proton fusion is doable. For a type III civilization capable of destroying starts that's also tiny. Remember your suggesting dysons spheres as the 'reasonable' option.
PS: The proton proton chain also produces deuterium as an intermediary. https://en.wikipedia.org/wiki/Proton%E2%80%93proton_chain_re...
There are huge number of assumptions based on that idea. A fundamental one is that a civilization in a relativistic universe would grow to densely inhabit a galaxy. Another is that they would need that much energy. Another is that even if they did have need for that much energy, they'd bother collecting EM radiation from stars.
That third is an interesting one to me. Now, you could muck up the night sky for the entire Milky Way, or you could, with Sufficiently Advanced Technology (tm), annihilate a total of 10 Jupiter-masses a year across the whole galaxy. For your average star system, that's extracting energy from 1.1 billion kg of mass a year, or roughly what NYC's garbage collection services dispose of every two weeks.
Pondering that, I suspect it's the wrong idea to imagine that a very advanced civilization, something we can only understand in the dimmest sense, would be constrained by something as specific as stellar energy emissions.