The story is almost certainly “The Three Tools of Death”, a Father Brown mystery, in which Sir Aaron Armstrong appears to have been murdered. A scene is set where it appears he has been shot at, stabbed and an attempt made to hang him before he was pushed from a window.
Fr Brown determines that Sir Armstrong was actually in the grip of a suicidal mania, triggered by breaking his temperance vow as a result of being driven to the edge of sanity by his "religion of cheerfulness".
The opening of the story introduces Sir Armstrong:
For though Sir Aaron was a philanthropist, and thus dealt with the darker side of our society, he prided himself on dealing with it in the brightest possible style. His political and social speeches were cataracts of anecdotes and "loud laughter"; his bodily health was of a bursting sort; his ethics were all optimism; and he dealt with the Drink problem (his favourite topic) with that immortal or even monotonous gaiety which is so often a mark of the prosperous total abstainer.
The established story of his conversion was familiar on the more puritanic platforms and pulpits, how he had been, when only a boy, drawn away from Scotch theology to Scotch whisky, and how he had risen out of both and become (as he modestly put it) what he was.
Yet his wide white beard, cherubic face, and sparkling spectacles, at
the numberless dinners and congresses where they appeared, made it
hard to believe, somehow, that he had ever been anything so morbid as
either a dram-drinker or a Calvinist. He was, one felt, the most
seriously merry of all the sons of men.
The story does not quite match the description of requiring all around him to be cheerful, but his servant does say
"My poor old master made game of me for wearing black; but I always said I should be ready for his funeral."
Thank you to @Randal'Thor for finding the full text at wikisource