That's strange on so many levels.
1. What do they mean by "Nothing we could do about that". I have some experience working in big multicontinental companies and they handle this kind of stuff every day. There are special people whos specialty is to prevent false accusations of the employees, stupid rumors and stuff like that
2. I didn't get the part where you tell us that you will not get the reference. If the CEO did not believe that it was you why did he not give you a good credential?
3. Maybe this is just a stupid rumor that started from nothing and goes nowhere (these things happen every time and in the end, everyone is forgetting about them ) It did not sound like they started a self-investigation ( what they should do by the way )
4. If you worked for the company for so long why there is no one to back you up?
I the end everything that I have to say is that you can totally use some online services like https://resumecvwriter.com/free-resume-review to help you with your resume. I know it not exactly what you are looking for right now but you probably need all the help you can get. Good luck man
Ahahaha Maybe they did not read to that part. I think no one reads past your job experience so you may write just about everything down below and no one will bat an eye. I also recommend that use some tools to check your resume, there is whole bunch of them, https://resumecvwriter.com/free-resume-review for example
I the end everything that I have to say is that you can totally use some online services like https://resumecvwriter.com/free-resume-review to help you with your resume. I know it not exactly what you are looking for right now but you probably need all the help you can get. Good luck man