In the realism school of thought, there is the concept of raison d’etat (national interest) which was heavily developed by Cardinal Richelieu. Cardinal Richelieu believed that the state needs were not based on the desires of its government, personal desires or universal demands of religion but raison d’etat. During the Thirty Years’ War, Cardinal Richelieu sided France against the Pope and the Catholic League and supported the Protestant princes. Raison d’etat explains this act by showing that supporting the Protestant princes would weaken Central Europe through division and erase the threat of a united Central Europe conquering France
However, Cardinal Richelieu still defied the Pope and sided with "heretics". Why was he not excommunicated for this?
Edit: I found this thread on here with the same question and only one reply which in sum states:
I see it as a possible theory, however, I will keep this up in hopes of something more substantive.
Lucien Bély has a good explanation for this in his book on XVIIth Century France (reference bellow). You are right to say that the "raison d'État" was the justification to side with Protestant states.
In 1625, two pamphlets against Richelieu were published : Mysteria politica and Admonitio ad regem. Bély writes that the former "[blames the cardinal for pushing the King into ruining Catholicism in favor of the Protestant party by helping Protestant states against Catholic ones - the Dutch against Spain, the Palatinate against Bavaria*]".
Friends of Richelieu would answer in two books : Fancan's Miroir du temps passé à l’usage du présent and Jérémie Ferrier's Le Catholique d’État. On this last book, historian Sylvio Franceschi writes : "[Distinguishing politics and the Church, the secular and the spiritual, Ferrier formulates, under the influence of Machiavelli, a theory that strictly separates civic morals and religious principles*]".
As Bély writes, Richelieu could rely on old arguments justifying being on good terms with the Ottoman Empire and heathen lands. "[Examples from the Old Testament and the History of the early Christian times served as justification for a politic that was meant to defend the Kingdom against foreign ambitions*]".
Richelieu could also rely on a long tradition of theological debates about the influence of Rome in France. As Guillaume Poumarède argues, it was, after all, "[an old quarrel between Church scholars that could be warmed up from time to time*]".
In brief, as a man of the Church, Richelieu was quite good at justifying raison d'État by relying on theological arguments. Why then would the Pope intervene against the largest Catholic country in Europe ? Still, at home, many denounced Richelieu's hypocrisy - laying siege to La Rochelle against the huguenots while shaking hands with Protestant princes abroad.
*Translations mine
Bély, Lucien. « Chapitre VII. Richelieu, Premier ministre », , La France au XVIIe siècle. Puissance de l'État, contrôle de la société, sous la direction de Bély Lucien. Presses Universitaires de France, 2009, pp. 167-208. (available online)
Franceschi S. de, « La genèse française du catholicisme d’État et son aboutissement au début du ministériat de Richelieu. Les catholiques zélés à l’épreuve de l’affaire Santarelli et la clôture de la controverse autour du pouvoir pontifical au temporel (1626-1627) », Annuaire-bulletin de la Société de l’histoire de France. Année 2001, Paris, 2003, p. 19-63. (available online)
Poumarède J., Thomas J. (sous la dir. de), Les parlements de province : pouvoirs, justice et société du xve au xviiie siècle, Toulouse, 1996.
I asked a different question about France joining the war, which was answered by u/Itsalrightwithme . You might be interested, as it covers a little bit of the ground of your question, and the Pope is mentioned in passing.