Overall I think the main agreement here is that humanity is far too ignorant to even posit how a type III would operate - never mind figure out how to detect them.

Stars use different fuel than reactors. To fuse simple hydrogen, four nuclei have to collide at once. This is so unlikely that it requires a very large mass of fuel to sustain a reaction: at least the size of the Earth, according to one article I saw.

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.")

That's not accurate, stars have several fuel cycles for pure hydrogen. At 1.3 solar masses you sun mostly get: https://en.wikipedia.org/wiki/CNO_cycle

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...