Why is The Power of Fables, in Book VIII of Fables by La Fontaine, addressed to Paul Barillon, the French ambassador to England?

The OP writes that "La Fontaine was a critic of French society to say the least". Sweetser describes him as "a loyal but lucid subject of a king eager to concentrate the power and the glory in his own person" (page 135).

Sweetser goes on to say that

The individual men and women praised by free spirits like Théophile and La Fontaine were patrons with whom they felt a personal bond, (...). Always aware of the taste of his public, the poet managed to renew a genre often very circumscribed by its conventions by seeking subtle correspondence between his topic and the traits and qualities of his dedicatees. Thus, when he dedicated "Le Pouvoir des Fables" to his friend, Paul de Barillon, French ambassador to the court of Charles II of England, he links the power of the fables that can influence people through the spell they cast on their listeners to the power of persuasion that a skillful envoy can exert on a sovereign whose friendship and support are necessary to the French king in order to ensure peace. He avoids an emphatic and pompous encomium, invoking M. de Barillon's modesty, characteristic of the perfect gentleman, to stress what is vital in his view for the good of France and its people: war should be avoided at all costs, peace should be saved through diplomacy.

Regnier does not say why La Fontaine dedicated "Le Pouvoir des Fable" to Barillon but notes that the ambassador did not foresee the Glorious Revolution of 1688 even though the count d'Avaux, ambassador in Holland, had predicted it every day. However, "Le Pouvoir des Fable" was published in 1678, ten years before the Glorious Revolution, so Barillon's lack of political foresight does not appear to be relevant here.

Regnier identifies several connections between the dedication and the fables, i.e.

• between line 11 ("Toute l'Europe sur les bras") and Fable XIV in Book VII, "L’ingratitude & l’injustice des hommes envers la Fortune" (Fable XIII on Wikisource but Fable XIV in Wright's translation and in Regnier's edition);
• between line 11 ("Toute l'Europe sur les bras") and the epilogue Fable XVIII in book VII, "Un animal dans la lune" (Fable XVII on Wikisource but Fable XVIII in Elizur Wright's translation) and in Regnier's edition);
• between lines 31-32 ("Sur les éloges que l'envie / Doit avouer qui vous sont dus") and lines Fable II in Book VII, "Le mal marié" (Fable II on Wikisource and in Wright's translation); this is merely a grammatical similarity, however.

Louis XIV needed an ambassador at the court of Charles II. Charles Athanase Walckenaer notes in Histoire de la vie et des ouvrages de J. De La Fontaine. Tome Premier,

Pour ces négociations délicates, Louis XIV choisit Barillon, homme d'un esprit vis, aimable, ami intime de madama de Sévigné, de madame de Grignan, sa fille, de madame de Coulanges, et de toute la société que la Fontaine fréquentait le plus habituellement, et où il se plaisait davantage.

Jean-Marc Bassetti notes in a footnote on his website that regularly visited the salons in Paris, including Mme de la Sablière's, where La Fontaine met him.

(There are plenty of sources that mention in a footnote that La Fontaine dedicated "Le Pouvoir des Fables" to Barillon without mentioning why; see for example in the letters of Madame de Sévigné, the letters of Madame de Maintenon and the Memoirs of de La Rochefoucauld.)

References:

• Oeuvres de Jean de la Fontaine. Volume 2. Edited by Henri Regnier. Hachette, 1884.
• Sweetser, Marie-Odile: "The Art of Praise from Malherbe to La Fontaine". In: Birberick, Anne Lynn; Ganim, Russell (editors): The Shape of Change: Essays in Early Modern Literature and La Fontaine in Honor of David Lee Rubin. Rodopi, 2002.
• Walckenaer; Charles Athanase: Histoire de la vie et des ouvrages de J. De La Fontaine. Tome Premier. Librairie de Firmin Didot Freres, Fils et Cie, 1858.