“The Bourbons”, or even only “the French Bourbons”, means a lot of people covering a lot of time, with a lot of differing attitudes towards religion; and a few of them even had much-varying attitudes. Of course the attitudes of Henri IV (1553-1610) and his grand^(6)-son Charles X (1757-1836) had no intrinsic reason to be the same!

You are right that Henri IV is famous for having converted 6 times back and forth (he was baptized, and died, as a Catholic) according to the needs of the moment. All the subsequent Bourbons were at least nominally Catholic. However, his son Louis XIII took the Protestant side in the Thirty Years War and (minimally) upheld the edict of Nantes tolerating the Calvinist religion in France (among other things: this edict also granted them their own fortified cities, which was kind of a political problem).

For later Bourbons, this picture changes a bit: Louis XIV, while not too much interested in religion in his youth, took a turn towards strong Catholicism later in his life, possibly due to the influence of his morganatic second wife Madame de Maintenon. The most famous example of this is the Edict of Fontainebleau, which ended whatever religious tolerance remained of the Edict of Nantes (although this edict had been slowly dismantled over almost one century, mostly because of its political impoications). Louis XV renewed persecution of Calvinists by the declaration of 1724, while Louis XVI was more tolerant (he officially recognized Reformed religion, as well as Judaism, in 1787) (yes, the Revolution guillotined the wrong Louis, this was a relatively decent Bourbon...).

After the Revolution, of course the attitudes of the later Bourbons were very much shaped by the Revolution itself; the shock of having guillotined Louis XVI pushed pretty much all of them to the fundamentalist side (moreover, the kind of people who still supported Bourbons in the 19th century tended to be fundamentalists already). The last two French Bourbons (Louis XVI's brothers, Louis XVIII and Charles X) were much closer to the Church (for example, Charles X was toppled in part because of pushing a law imposing the death penalty for profaning the Eucharist — this was already completely anachronic by 1830).

There are a number of considerations missing from your question/post.

First is whether Henry IV said "Paris is well worth a mass." Historians consider the alleged quip to be much more likely apocryphal.

Second is the view on religion at the time, whether Henry IV's conversion to Catholicism was political and not spiritual. One can question Henry IV's theological depth, but he consistently displayed sincerity in his abjuration of Protestantism and embrace of Catholicism. Just to give a few examples, he regularly attended Catholic service, in many places in Paris and beyond. He took part in religious processions throughout his reign. Further, he attempted to lead other (former) Huguenots to convert to Catholicism, including his own sister Catherine de Bourbon, princess of Bar, by assigning his personal confessor the Jesuit Father Coton to her household.

Third is whether he advanced policies consistent with his faith in Catholicism -- or lack thereof. Many would cite the Edict of Nantes as part of Henry's alleged stratagem of "pretending to be Catholic while advancing Huguenot causes." However, this is clearly not true, as the Edict of Nantes strongly benefited Catholics by restoring Catholic services and privileges everywhere while Huguenots are curtailed.

Digging deeper, one may wonder why Huguenots were given strong forts such as La Rochelle to keep as a refuge. To understand this, one must understand that at this phase of the French Wars of Religion, there was general exhaustion on all sides, and rising discontent. In particular, peasant groups had started to arm themselves in the countryside -- called Tard-Avis'es -- to the point that neither the Catholic League nor the Huguenots nor Royalists could safely operate in many areas. So if he were to continue fighting, he risked a descent into civil disorder and breakdown of authority.

Tellingly, the articles of the Edict were quite punishing to Huguenots, for example in order to retain La Rochelle they had to pay the king a large annual sum. Even so, he could not appease everybody. In particular, he had had to fight harder for the Protestants in order to come up with an acceptable settlement.

In addition, the Edict of Nantes is often portrayed as being intended to be a permanent document of toleration. This is clearly not the case, as the preamble itself states:

But now that it hath pleased God to give us a beginning of enjoying some Rest, we think we cannot imploy ourself better, to apply to that which may tend to the glory and service of his holy name, and to provide that he may be adored and prayed unto by all our Subjects: and if it hath not yet pleased him to permit it to be in one and the same form of Religion, that it may at the least be with one and the same intention ... and that we and this Kingdom may alwayes conserve the glorious title of most Christian, which hath been by so much merit so long since acquired.

Finally, we must go back to the subject of Protestant possession of fortified towns such as La Rochelle. This part of the settlement was not part of the Edict itself, which required registration by Parliaments. Rather, they were part of two brevet articles, granted by the king himself, only for a period of 8 years. Therefore, it is clear that Henry IV intended for this appeasement to be temporary.