It sounds like Keith is more interested in one upping his coaches. You are trying to get the guy to self incriminate. If the words warranted an ejection, then you don't need the self incrimination. Reffing isn't a game. It's a profession.
Great refs understand that managing a difficult coach requires great skill.
They aren't interested in revenge or showing the coach up. They are more interested in protecting the ongoing game using all their formidable skills and tools. The most effective refs that I worked with used a combination of humor and authority to difuse the situation. I suppose there a lots of possibilities. It would be better to ask him if what he said meant he thought that you were cheating.