If you convert:
5 / 16 inches = 7.9375 millimeters
0.0625mm is scarcely 2 and a half thousands of an inch — I’ve been using 8mm and 5/16" tooling and collets pretty much interchangeably for a while now — granted I don’t do terribly fine work, but it should work for your needs if the length isn’t too excessive.
Certainly @RichCournoyer 's concerns about safety should be taken seriously.