I would like to know how a native speaker would say that someone tries to smile, however hard it may be for this person to do so (because he is embarrassed, or because he simply is reluctant to smile at something which he does not find particularly funny). It seems that "he endeavoured to smile" is not quite right (only five results on Google Books, only four with the American spelling)... What could one say, then, to express such a thing, insisting on the effort which the person has to make to smile?
I think you've picked the perfect word for what you're trying to express.
Here's a definition that matches this usage fairly well, from the Oxford Dictionaries:
Earnest and industrious effort, especially when sustained over a period of time
You could say "struggled", but that might imply that he's not succeeding.
You could say that he "forced a smile". I think that implies the smile looks a bit fake, and probably didn't last very long. If they kept up a smile over the course of a dinner, that would be "endeavoring", if they gave a quick smile when greeting someone, that might be "forcing" it.
Just because a specific phrase doesn't come up in a search doesn't mean it's incorrect, but of course it doesn't hurt to check.