British Peerage in Late Elizabethan England

Since he's a privateer, I'd think "Captain Lastname" would do.

Marlys is right, but you could have him made a Baronet, which is equivalent to being a hereditary knight. He would be addresed as Sir whatever. https://en.wikipedia.org/wiki/Baronet

If he's a younger son of an earl or the son of a viscount or baron, then he's plain Mr. Firstname Lastname, so that might be the way to go (any higher and he'd have a courtesy title of Lord Firstname Lastname.) And yeah, Captain would work.

Peers generally refer to themselves by their last name only, however bear in mind that such a practice would mark themselves as a peer. For example, Lord Tukesbury would refer to himself and sign his name as Tukesbury. Other people would refer to him as Lord Tukesbury. Non peers would be Mr (last name), to themselves and others.

The practice of referring to people by their last name rather than first name is (or at least was) quite common among upper class and upper middle class people. Public* school children have generally been referred to by their last name only (see the whole "Jennings" series set in 1950s boarding schools). I don't know if this is still the practice in these schools but it always used to be like that.

*by which I mean places like Eton, i.e. highly exclusive private schools. Government funded schools in the UK are called state schools.

In other words, I'd see no plausibility issues whatsoever if an upper class character was referred to by their last name only throughout the entire book. His immediate family would use his first name though - if his first name has to be hidden and he has conversations with his immediate family, then you may need to be a bit careful here, but he might be known by a nickname (especially if his first name is the same as his father's, as is common for eldest sons, for example).

And just for comparison, in the above mentioned Jennings series, Jennings's first name is almost never mentioned. I think his mother uses it when writing to him. I don't even remember what his first name is, in spite of having read lots of Jennings stories.

In the scenario you describe, I'd choose the nickname or "by-name" route. You could also set it up so that his Christian name is so little known outside of his family, that your listening character could be heard to say, "Sir WHO Douglas" or whatever, since he knows your Main Char by his by-name exclusively. He could be known as Four-Finger Douglas or Bit-Cat Douglas or whatever, to the social exclusion of his Christian name. Even today, I have a brother whose given name is so seldom used, he has to put his nickname on his business cards. We in the family know it, of course, but very few outside, and we never call him by his given name.