What is the closest mm size screw to 5/16\" ?
The closest mm size screw to 5/16" is a 8 mm screw.
I hope this answer is good for you.
Proposal:
I used to work in a small factory in Romania, actually i was in charge of almost everything there, suppliers, subcontractors. I remained in good relationship with them, so i already know a few suppliers for you. I know a family business, a small factory that makes screws like that, even smaller, they used to make bolts and nuts for a company that produced fishing tools. They are very serious and strict about quality and deadlines.
I think that they can be a good fit for you.