Letterhead is optional. It's not even necessary for your recommender to be a current colleague comma so if you think about it that way definitely not.
I think it's a little bit strange that this person is treating a letterhead like it's a pot of gold or something. I mean it's a piece of paper. I've had co-workers at all levels of the organization write recommendations for me on letterhead.
Maybe you could tell this person there are insolent people all around the world writing letters of recommendation on letterhead as we speak. It's happening.
My greater concern in this situation is that your recommender might be pushing back on you. If they are in any way hesitant or seem to be trying to squirm out of helping you, definitely take this a different direction. Ask someone else.
There is absolutely no point to trying to drag someone through the dirt to be your recommender.
Farrell Dyan Hehn, MBA
Admissions Consultant & Verbal Strategist MBAPrepCoach.com