My answer will pertain to the Royal Navy around the time you specified, so the late 1700s leading up to the Regency era. In short, it would be very difficult for someone who came from a humble background to reach or even move past the position of lieutenant, though it was not unheard of. Proving oneself in battle, making important connections with higher-ups, or simply remaining in service long enough were ways for any sailor to be promoted, regardless of background.  

As for a more in-depth explanation:

In terms of the social backgrounds of naval officers during this period, only about one fifth actually came from the landed gentry/aristocracy (i.e. gentlemen), with most of course being younger sons who were in need of a career. A second fifth was made up of the sons of men already in the navy, while the remainder had fathers whose careers were in the fields of commerce or the clergy. Very few came from humble backgrounds, for example the sons of farmers, domestic servants, or in your case, a lighthouse keeper. 

Boys usually entered the navy at around 13 or 14 years of age, and began as a midshipman or a servant to the captain. It was up to the captain of a ship to approve the appointment of a midshipman, and he would usually reserve positions for younger relatives, the sons of friends, or any boy whose family was somehow connected to him. Networking was extremely vital, hence why it was rare for lower-class sons to find their way onto a ship, save for those whose families came from coastal towns or cities with naval bases. Thus, the son of a lighthouse keeper may have a better chance at a naval career. 

When it came to promotion, you couldn’t purchase a better commission like you would in the army — midshipmen had to wait about six years before they could take their lieutenant’s exam (during which they would the learn manual and technical skills of their profession) and after that they had to wait even longer to receive their commission and fill a vacant position (this could take days, months, or years). 

The rank of master (which was about equivalent to that of a lieutenant) was reserved for men who were experts in the field of navigation. This required them to be well educated and have a solid understanding of mathematics. Lower-class sons were thus almost always excluded from such a position, unless they were provided an education at an early age and spent their years as a midshipman strengthening their knowledge in this area. The rank of lieutenant was thus easier to attain, for it was not a specialized area. 

Once someone reached the position of lieutenant, it was uncommon to secure the rank of commander, and even rarer to snag a captaincy. Even most gentlemen never moved beyond this point, unless their family had considerable influence over senior naval officers/officials in the Admiralty, or if they proved themselves extremely capable at their job. 

Distinguishing oneself in battle was one way anyone could be automatically promoted. For example, John Hindmarsh, the grandson of a gardener, was a midshipman on the Bellerophon at the Battle of the Nile. During a violent encounter with a French flagship he soon became one of the last men standing on the quarterdeck, and realized the enemy L’Orient had caught fire. He ordered for the cable to be cut (the Bellerophon was at this point anchored) and set the spritsail, allowing his ship to sail away before the L’Orient blew up. As he was only 14 at the time, he could not be promoted, but received his lieutenant’s commission just four years later, became a commander in 1815, and eventually the first Governor of South Australia in the 1830s. This was a rare success story for a man of his background. 

For those who did not have any naval connections and did not see battle, their outlook was quite miserable. On average, lieutenants had to wait between 20 and 30 years to achieve the rank of commander, and this was often granted right before retirement. It took another 10 to 15 years for one to be made a captain, and at this point they’d be lucky to have energy enough to embrace this position. This route, though miserable, would likely be one a lighthouse keeper’s son would take. 

In the end, it wasn’t that lower class men couldn’t move beyond the rank of lieutenant, it was that most didn’t have good connections, an education, or even luck enough to enter the navy in the first place. But if the son of a lighthouse keeper did manage to find himself on a ship, he better work hard and prove himself worthy if he wanted to attain the rank of captain. 

Sources:

Charles Consolvo (2005) The Prospects and Promotion of British Naval Officers 1793–1815, The Mariner's Mirror, 91:2, 137-159, DOI: 10.1080/00253359.2005.10656942

Wilson, Evan. A Social History of British Naval Officers, 1775-1815. Woodbridge, Suffolk, UK; Rochester, NY, USA: Boydell & Brewer, 2017. doi:10.7722/j.ctt1kgqswg.

Muir, Rory. Gentlemen of Uncertain Fortune: How Younger Sons Made Their Way in Jane Austen's England. New Haven; London: Yale University Press, 2019