Tag with Aspects
Remember NPCs can create advantages, compel aspects, etc. If someone's acting in a very odd way it's likely that other characters can start tagging him with aspects disadvantageous to him. Someone that's doing everything flashy could easily get a "Showing off" situational aspect created on them that makes it easier for people acting sneakily or carefully to beat them in a contest (user can invoke advantage for +2)
That said: Change the Contests Away from their Advantage
When someone defends, the ways they can reasonably go about defending depends on how you attack. Someone has a harder time defending carefully if you're attacking quickly or sneakily, since it takes time to be careful. Similarly, avoid attacking a flashy opponent forcefully or sneakily, since these are the more reasonable times when they could use their preferred approach. Instead: be quick, be clever (the sample text for clever even says "finding a weakness in the opponents swordsmanship")
As a GM, you have every aspect and approach arguing power the player has, and an easier time approving your reasoning since it falls to the GM to approve approaches.
Of course this part relies on you not getting taken out before you have a chance to act, but hopefully your major villain can't be gibbed in a single action, give them stunts to act first or otherwise improve their survivability if you keep getting taken out before you get a chance to act.
Ultimately : Loose rules systems thrive or fail on the resolve of the GM
Basically, FAE's rules can be pushed any which way by someone that talks cleverly or imploringly enough, as the GM though you get to say though whether some reasoning is good or not. Same thing when it comes to aspects and stunts. As you noted, the game can't withstand it very well if someone can use +3 on everything. And if the other players aren't playing that way, it diminishes its value to the game. So you have to say no some times.
From the book :
So your first instinct is probably to pick the action that gives you the
greatest bonus, right? But it doesn’t work like that. You have to base your
choice of approach on the description of your action, and you can’t describe
an action that doesn’t make any sense.
Roleplaying is about keeping things fun and interesting. I've seen sessions be interesting and enjoyable laughfests because everyone was on-board with figuring out how to best solve problems according to their character's styles. If fun is being had, let them be a little crazy (and join in on the crazy with your NPCs), if things aren't being fun, it's time to put the foot down.