captain january,perhaps? shirley temple movie,1936,i think?
It does sound like "Captain January," It has been awhile since I've seen it on TV, but it was filmed in black and white and I vaguely remember a storm towards the end, but I don't remember a cat in it, and it was definitely not a Disney film, as the Walt Disney company did not make full length feature films till 1937, when they made the animated "Snow White and the 7 Dwarves." Live action films coming later.
The only thing I can remember with the Disney name attached and featuring a lighthouse was an episode of "Disneyland" called "The Mystery in Dracula's Castle" in 1973, which was in two parts, but was filmed in color.
Maybe someone has some more suggestions.