This is probably the last star you would want your Dyson sphere on.
I wonder if it would be something to do with neutron stars or magnetars?
Theoretically, you could place a relatively small, dense object fairly close to Betelgeuse and protect your entire stellar neighborhood. I haven't done the math, but I suspect it would have to be so small and dense that it would be a black hole.
You'd probably be better off just trying to shield your planet from whatever lesser energies eventually arrive.
A back cover blurb I wouldn't mind picking up! If something like this is already written I want to know.