I would, but our run isn't secure either. Its not predator proof and only able to with stand our day predators like dogs and hawks. We lock all of our chickens up at night in predator proof coops which would become to small if I put the brooder in there. If I put it in the run it would also have to be secure from other predators. Coons, Coyotes, Weasels etc.
That's probably a no-go then.
I made my brooder as predator proof as I could get it short of adding electric wire, but I couldn't be 100% sure that a large and determined dog couldn't tip it over.