You will need to use a 25/64" (I use cobalt as it will cut the best hole) drill bit to successfully tap a 7/16" x 20 hole. I know it's only a 64th of an inch larger than a 3/8" but it makes all the difference. Use a high quality bottoming tap with lots of thread cutting oil and go very slow as breaking off the tap brings a whole new set of problems.

Good Luck!
Glenn