I had heard that 'fag' was more likely to be misconstrued in the USA.
These days most Americans probably wouldn't know what you were talking about if you used that word. (Fag.) Twenty years ago maybe.
'Fag' is still a common extreme pejorative in the US, although it's probably more likely to be considered hate speech nowadays. Few Americans would know the BrE sense of the word unless they've had exposure to BrE culture or media. Ditto for the longer version 'faggot'.
I first learned the BrE meaning one summer when I worked with some Scottish guys at a residential summer camp. I was listening to one of them tell a story about getting caught smoking on camp grounds (a major no-no and grounds for potential dismissal). When I asked him how he managed to get caught, he started telling how the director walked around the corner and caught him "with a fag hanging out of his mouth".
Knowing only the AmE sense, I was of course very confused as to what exactly he was caught doing. He was similarly confused as to what else I thought it could possibly mean, being completely unaware of the AmE sense.
This was the same summer that the young British women working in the front office saw the director walk by, and ran out asking him loudly in front of everyone if he had a 'rubber' she could use at the front desk. Amid the howls of laughter from all the Americans, the red-faced director responded with "I think you may mean 'eraser' - otherwise I'm afraid I don't." She was mortified when we explained that 'rubber' was AmE slang for 'condom'. Someone even went so far as to leave a box of condoms at her desk later on.
Good memories.