Assuming that this takes place in the 17-18th century, it is important to note here that your hypothetical French nobleman should be absolutely comfortable with slavery if he planned to make a living in the French Caribbean, which owed its prosperity to the production of sugar and other commodities obtained using slave labour. The entire economy revolved around bringing African people to the islands, making them work in exacting conditions, and then bringing new ones. As someone of a certain status, your French nobleman, even if he was not a planter, would buy slaves, own slaves, etc. as a willing participant in the slavery-based economy.

The idea of a impoverished French nobleman going to the Caribbean to find a new life for him is a sound one. This happened often, and part of the ruling class included noblemen whose ancestors (or themselves) had left an unsatisfying life in France for the warmer and more promising pastures in the Caribbean. However, the path to riches usually included time spent in the military: once in the Caribbean, officers of the Royal Army found that they'd rather be planters than soldiers. This was particularly attractive for men of minor noble families, whose military prospects in France were limited (see u/waldo672's answer about the rank system in the Ancien Régime).

This was the case for instance of Alexandre Dumas's uncle, Charles Davy de la Pailleterie, who came from minor nobility in Normandy. After a short career in France, Charles was posted at 16 in a colonial regiment in Saint-Domingue. The next step typically involved marrying a woman from a planter family. The bride's background could be obscure but she had wealth, while the potential husband had a name. At 22, Charles thus married the daughter of a planter, and eventually bought his stepmother's share of the plantation, his Caribbean estates now dwarfing those in Normandy (Reiss, 2012). Another example of this scenario was the story of Joseph de Gripière, a man from a destitute noble family in Southern France. Posted in Saint-Domingue as an aide-major when he was 26-year-old, he was made a colonel at 31 and was rich enough to buy a large coffee plantation in association with a local carpenter. His marriage to a rich widow in 1780 made him a millionaire, owner of several plantations with hundreds of slaves. Another example would be the Lobit brothers from Gascogne, who were so destitute that they had to beg to find passage to Cayenne. Three of these poor aristocrats ended up in Saint-Domingue instead, where they made good marriages and became the owners of coffee plantations (Figeac, 2013).

Was buying a small ship and becoming a captain specialized in coastal or inter-island trade (cabotage) a possibility for a nobleman? Assuming that your nobleman wanted to command the ship himself (rather than just buying it and hire a crew), captaincy was not something that could be improvised, as Stede Bonnet found out: years of schooling and training were necessary until you became proficient enough to be a captain and earn the respect of your crews and the trust of your customers. Investing in a ship would be a better option for a non-seafaring person with money, but still a risky one: ships were expensive and the business was dangerous. It was also a sector that was difficult to penetrate: ship-owning and commerce in the French Caribbean was dominated by a handful of metropolitan families and their powerful representatives (sometimes family members) in the ports.

Was carrying mail a potential business? The concept of "packet-boat" (paquebot in French, paquebote in Spanish), a small ship that carried letters and passengers on a fixed route with a more or less regular schedule, had been implemented in the British Isles in the 17th century, and it was also known in France, where a paquebot was crossing the Channel from Calais to Dover (Monconys, 1677). In the Americas, as early as 1525, the Spanish had created a system of dispatch boats exclusively dedicated to the transportation of post in the West Indies, though it had functioned poorly. In 1764, this system was partly replaced with a Maritime Post system (Correos Marítimos) that carried post, freight, and passengers on a monthly basis from Spain to Havana with subsidary routes to the viceroyalties of Mexico, New Granada, and Peru. Note that to be hired as a captain, one had to have sailed for six years at least as a sailor, and two as a pilot, which itself required a 6-month training in navigation school and several training voyages (Baudot Monroy, 2015).

In the French Caribbean, however, mail was still transported by regular ships of the naval and merchant fleets. Ideally, the captain was given a bag or crate with letters and parcels collected at the ship-owner's office at the port of departure, and he delivered it to the ship-owner's office on arrival. In practice, the system was less strict and hardly satisfying, as letters were often given to crew members who lost or damaged them. There was also a global lack of privacy and secrecy: nothing prevented a merchant present in the office from opening a letter containing sensitive information addressed to a competitor. In the second half of the of 18th century, public and private initiatives tried to set up more efficient paquebot systems, but it does not seem that they were successful, due to the opposition of local or metropolitan stakeholders who were alway wary of monopolies. In any case, setting up reliable fixed routes across the Atlantic and in the Caribbean remained challenging in the 18th century (Charriaut, 1951; Jolivet, 2021).

Going into the general cabotage trade could be a more valuable idea, as short-distance trade was important for the economies of the French Caribbean. Its main purpose was to carry goods between the lesser harbours and the main ones. It made possible to export colonial products to France - sugar from Guadeloupe, for instance, was brougth to Saint-Pierre in Martinique) - and to import to all the islands the products brought on French ships. Cabotage was also fundamental in circumventing the Exclusif principle, which forbade French colonies to trade with other nations. Because it restricted imports to those brought from France, the Exclusif limited the availability in the islands of certain goods, notably the dried fish and meat needed to feed the slaves, and the availability of enslaved people themselves. This interlope (contraband) trade allowed a freer circulation of goods throughout the Caribbean area and to mainland America. In the late 1700s, the cabotage was carried out by small crafts, such as local schooners (goélette) called balaous. This was a booming business: according to Pérotin-Dumon (1991), the local trading fleet of Guadeloupe in 1770 had three hundred to four hundred sailing masters, most of them "poor whites", corresponding to 150 to 200 boats each with a crew of four slaves. Wars and cataclysms (such as earthquakes) made this type of business even more prosperous: the main official supply routes were disturbed, making the informal ones even more necessary to feed the populations. Local caboteurs could also turn to privateering. There are some passing references to caboteur captains but information seems limited, perhaps due to the fact that this trade was partly illegal. Much of the caboteur trade was in the hands of foreign powers anyway (Frostin, 1967).

Conclusion

Destitute French noblemen did go to the French Caribbean to find a better life. The "easiest" path was to leverage the social capital that came with the aristocratic status: obtaining a commission, becoming a member of the militia, having an attractive "name". No only this would put some cash in your pocket, but you could use your newfound prestige to rise in the creole high society, where many people were willing to exchange money for a name. A good marriage could get you a plantation, resulting in more wealth if one managed the plantation properly (or in a lot of debt to merchant houses if you did not, which happened to a few aristocrat planters). Then, perhaps, if you had the right connections, you could invest your money in trade and ship-owning ventures.