Trump is not trained as a politician. He has a great many talents, but deflecting the painful rhetoric is something he is having difficulty with.
I have seen pics of him with the queen and smiling, but he is remaining serious most of the time.
Donald has said over several years that he has deep respect for the queen. I imagine some of it is simply feeling awestruck, and focusing on her every word.
Last year they had a tea together. She liked that time very much. This is known, because she is very punctual. Also, she calls things short if she is having an unpleasant time. She stayed at that tea an extra seventeen minutes, which is very significant for her.
HRH is the type who, if he indicated a willingness, she would coach him on etiquette.
HRH is bulletproof. If anybody could teach the president how to become immune to the hate, it would be her.
With American press being plenty happy to post unflattering pictures of Trump, your best bet for finding a smile user-generated the British press. They have a few favorable stills and video shots of him.
By the way, on one coverage of his visit they mentioned that the reason he declared that there were only a few protestors, is because his itinerary was intentionally set up so he would not see protests. He didn’t know they were there.