This question seems related to my question, but it does not have an answer. I hope to get an answer by asking something more specific.
I am a postdoc at a research-only institution whose only students are in doctoral programs. People are allowed to be (co-)appointed with other university and teach there, and some of them have taught, but none of my colleagues currently are to my best knowledge. I would like to have recent teaching experience (I have more legitimate teaching experience where I obtained my doctorate) and obtain a letter of recommendation regarding teaching.
To this end, I thought of holding a mock class, where I give a lecture on something that is taught in some typical undergraduate course to PhD students and senior researchers, one of whom could later write a letter to me.
My question is whether or not this way I can get teaching experience and a letter that are at least somewhat relevant in job applications to (say) American math departments. (Addendum: I'm not in the US right now.)