This is a valid concern as it’s a delicate subject to bring up with the current employer. I would also love to hear how others have handled this.
If I could add a related question: personally I have access to the official job description on company letterhead which matches my position exactly but without my name on it. I guess this is logical, it’s a job description describing the specific job not the employee. If I’m able to provide the job description and my offer letter (or a letter of employment verification), which will match up regarding the title, will that be considered sufficient?