To me it depends on so many things. The pov character, the setting, and the writing style of the fic are the main ones. If Francis has the pov focus then Iād call him by first name, unless Iām doing something stylised where itās about his role as captain, or his conception of himself in that role - his title is very important to him, and there are actions he takes as Francis, whoās just some guy, and actions he takes as Captain FRM Crozier of Her Majestyās Ship Terror. Proposing to Sophia is Francis, abandoning ship is Captain Crozier, for instance. So even within a fic there are different situations where it can make sense to refer to him by different names in different scenes.
(If youāre aping the book, I think heās mostly referred to as Crozier in that, or with his mile long full name; Francis Rawdon Moira Crozier. Which does give it quite a pompous air. I havenāt made it through that brick myself, so Iām mostly going off the show!)
If itās a James pov Iād almost always go with Francis for descriptions as well as his thoughts and/or internal monologue, but vary between that and Captain Crozier in dialogue depending on where their relationship is at - itās Francis when heās trying to befriend him, Captain Crozier when heās showing him respect, Francis when heās actively antagonising him (heās his subordinate, so using first names is an insult), Francis face to face and Captain Crozier to others when theyāre on good terms.
In general, people who are of a rank or superior would call him Francis in canon, as well as civilians with whom he has a close or intimate relationship. So if weāre seeing him through any of their eyes - eg, Franklin, Ross, Sophia, Lady Jane - thatās what heās called. His subordinates, however - Fitzjames and anyone below him in rank - would call him Captain Crozier or the Captain. The exception is Blanky; if heās the focus on the story itās first name basis.
If itās a modern AU, this all kind of goes out the window - 21st century guys are much more chill about addressing each other by first name, and calling him Crozier in that context implies much more of a distance between narrative and character.
The writing I know best is my own, so forgive me for using my own fics as examples, but in The Bus Stop on Terror Street I consistently called him Francis, because itās a modern AU from his point of view, where his role as captain is wholly absent and he doesnāt stand on ceremony as much as he does in canon. In Clean Cut, heās the Captain or Captain Crozier, because we are in canon era and the pov character is Little, who is his subordinate and respects the hierarchy. Itās not about Crozier, but as an example of style affecting name choice, in Mr George Hodgson I almost exclusively refer to George as either Mrs Irving or Mr George Hodgson because the writing style is very formal and the story is about the different identities those names imply. A simple āGeorgeā just wouldnāt achieve the same effect.
In summary, it depends on who he is in the story. Is he regular-guy Francis, is he the (dis)respected Captain of Terror, is he the impersonal last name Crozier, or is he our good buddy Frank? If heās mentioned a lot in the same scene you might want to use several different ones. Unless you want to make repetition a Thing, which is sometimes cool. There are no rules; itās writing.
