If we exclude the Dukes of Normandy and shorten the list to those who held an immediate claim to the French throne, then one candidate remains that I am aware of: Louis VIII aka Louis Le Lion.

In 1215, King John immediately attempted to renege on the terms of the Magna Carta. His recidivism prompted Baron Robert Fitzwalter to muster other major English landowners in an uprising against their monarch, commonly referred to as the First Baron's War. In need of both military support and a leader to effectively replace John, the barons sought out a suitable, powerful claimant for the throne. They found Louis le Lion, the eldest son of Philip II of France. Louis was a relatively well-regarded warrior and incredibly wealthy, with the ability to draw on significant resources to support the barons and stake his claim. Equally important, he was a direct descendent of William I, married to John's niece, and was actually willing to back a group of rebellious noblemen for a shot at the throne. With the tacit endorsement of his father, who may have disapproved of the venture from the start, he pressed his claim, raising a fleet and an army and sailing across the Channel in 1216.

Initially he met with nothing but good fortune. John retreated, a significant number of English nobles recognized his claim, and he captured both the essential cities of London and Winchester largely without opposition. A series of sieges and maneuvers followed (including the famous siege of Rochester Castle, ridiculously retold in the film Ironclad). Ultimately, Louis emerged in a solid position with a good chance of actually securing the crown. Unfortunately, John had the good sense to go and die of dysentery a few months after Louis' triumphant invasion. With the wildly unpopular king dead, English nobles were quick to flock to Henry III, his nine-year old son, as a more suitable (and more easily manipulated) candidate for the throne than the French prince they had so recently proclaimed as king.

The charismatic and highly respected William Marshal, Earl of Pembroke, made huge efforts to secure the loyalty of the English barons for the young Henry III, and eventually built up enough support to pursue a war against Louis. Not only that, but he leveraged his influence to convince the Pope to excommunicate the French prince. As the conflict wound on for about a year, Philip II eventually stopped offering his son any support for his English venture. With his fleet (under the command of the infamous pirate Eustace the Monk) smashed in a naval battle near Sandwich, his land forces roundly defeated outside Lincoln, and public opinion in England largely turning against him, Louis had little choice but to enter into negotiations to end the conflict. He signed the Treaty of Lambeth in 1217, part of which involved him agreeing that his claim to the throne was never legitimate in the first place, and returned to France in defeat.

Don't worry about Louis though, he did eventually become King of France, at which point in time he led a successful crusade against the resurgent Cathars in the south before dying of dysentery in 1226. Henry III went on to rule England for 56 years as a pious but relatively unimpressive monarch. He even got a cameo in Dante's Purgatorio, where he sits, slightly apart (probably due to his piety), from other failed rulers of Europe.

Sources:

https://yalebooksblog.co.uk/2016/06/02/forgotten-king-england-louis-viii/

Louis: The French Prince Who Invaded England by Catherine Hanley published 2016