Glad to be able to help.
Depending on what type of chuck is on your brace, you might have to file some flat spots on the sides of the shank of your new screwdriver bit. If it is just plain round I am afraid that it will slip when you try to tighten a large screw with it.
That’s why my idea was to use an old brace type drill bit. They have that conical square end that helps to ensure it doesn’t slip.