Sure that will work. People have been cooking with cast iron on all types of heat for years. Wash it out with dirt when through
Being the pack rat that I am, I have a small collection of toaster oven broiler pans. I use those for the grill, sometimes lining with foil. They're light enough to move around easily with the tongs. And I'm weird in that I like a special tool for a special purpose. The grill has dedicated tongs and spatulas, too.
One thing to watch for when you are done is the bottom of your skillet might be black.