Q. I need to find a way to apply a surface mirror coating.

These societies may help you find the chemical solution used to test the accuracy of a surface mirror before it's sent out to be coated with the evaporated aluminum. 

The Glass Association of North America may be your best bet: 
https://www.glasswebsite.com/default.asp 
Their technical section provides information about mirroring: 
https://www.glasswebsite.com/techcenter/Information/index.as
p#Mirrors 
They have a separate mirror division: https://www.mirrorlink.org/ and a list of members: https://www.mirrorlink.org/tech/links.htm 

Telescope mirrors require high-tech solutions to "silvering." The American Astronomical Society may be able to suggest an expert to help you:https://www.aas.org/ 

You may wish to contacting the American Scientific Glassblowing Society to find an expert who can help you with this project. Laboratory glass may be silvered and the glassblowers usually have the chemistry background to adapt their technology to your black glass spheres. https://www.asgs-glass.org/ 

Another useful resource is the Society of Glass and Ceramic Decorators (SGCD). Their website states the SGCD "...is a worldwide organization that provides decorating professionals with a competitive edge in business by providing opportunities for networking to learn about new decorating technologies and techniques." They may have an expert who can answer your questions as well. https://www.sgcd.org/