You ask a question that cannot be answered in accordance with the standards of this website, because there is no compendium to which reference may be made of what "a native English speaker" would say. The only reference would be what a large number of native speakers aver they would or would not say.
For my contribution to that compendium, I state that "I will put you in for Tuesday for 10 AM" sounds to me exactly what I would expect to hear, and have, in fact, heard from diary secretaries in Britain on many occasions.
Its tone is somewhat dictatorial, but so then are some diary secretaries.
When it was part of my job to make appointments for subordinates to see a very important person, I think I might have preferred to say "I will book you in for ..." but the tone would have been the same. I would not have used it if the person wanting to see my boss was the Pope, or the Queen of England.