Can't completely verify this.

First, the anecdote is for Francis I. The king told him to stop making fun of the ladies of the court ( and likely Francis' mistress). He didn't. The king condemned him to death, then asked how he would like to die. "My lord, by Saint Nitouche and Saint Pansard, patron saints of lunacy, I ask to die of old age" The king laughed, and commuted the death penalty to banishment.

There are many bon mots attributed to him. A good one is when Francis, enmeshed in the Italian Wars with Charles V, was meeting with his councilors planning an expedition to Italy. Triboulet, asked his opinion, pointed out that everyone was planning how to get into Italy, but that if they didn't want to stay there, no one was talking about the important thing: how to get out of Italy.

Francis indeed went into Italy... and indeed was captured after his defeat at Pavia the next year, in 1525, and so stayed longer than he liked. Was Triboulet really that prescient? Or were these mostly remarks just ascribed to him later ( like Yogi Barra, "I really didn't say everything I said") I don't know ( where's the Renaissance French Lit specialist?) but you can read some of them in M. A. Gazeau's classic Les Bouffons (1882) here at the Internet Archive, and still in print.

EDIT: the one about leaving Italy goes back at least to Jean François Dreux de Radier; so 1767. The one about dying of old age I have only tracked to this 1866 Dictionary of Biography, and though none of its cited references don't have the anecdote it's likely not the earliest mention of it.

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