I am certain that a GM who pays 4 fps to a character has very strong narrative reasons for doing this
Sorry but I disagree.
IMHO if the GM needs to
bribe the players to make them do what he wants, there's a problem in narration.
Perhaps he gave too few hints or traces, perhaps the dark alley in fiction hasn't became so important for PCs to go down it.
For this reason I'd like to know the whole story, just cause for now I'm just speculating.
In Fate and in RPGs in general you got so many ways to tease characters along the path you have chosen. And however you should expect characters act on their own.
In the latter case, you simply adjust the scenario on the fly.