Generally speaking the lighthouse keeper and his family didn't keep warm any differently than other people of the time period, no matter if they lived in the lighthouse proper or in an adjunct cottage. Iron cast wood stoves or bricked fireplaces were the way to go. I think masonry heaters like the were widely used in Germany never really caught on in the UK and US so it's partly also location dependent. Because heating is expensive, likely only one central room, the kitchen/living room, was heated while the bedrooms stayed cold and for his duties on top of the lighthouse the keeper would have to bundle up.

For example, the website for the Farne islands lighthouses has an article about famous lighthouse keeper's daughter Grace Darling and it mentions in an aside how the family lived:

The family spent most of their time on the ground floor of the lighthouse which comprised of a large room, heated by a wood stove. The room was their living room, dining room and kitchen in one and had a spiral staircase leading to three bedrooms above and of course the light at the top of the tower.

The website is gone but a snapshop of the site can be found here.

This was ca. 1826 in the UK so a bit later than your story is set, but it should still fit.

A depiction of the same lighthouse interior showing a scene in 1938 shows an open fireplace in the background, but since this picture was painted about 30 years after the fact it doesn't necessarily have to be accurate but it's not necessarily wrong either.