In my admittedly limited experience the big problem with single classed Fighters is their low Will save. That could be a problem from time to time in the AP you’ve selected, but there should also be a lot of fights it won’t factor into (unlike in a homebrew where the DM finds your weak spot and keeps hitting it over and over).
Other than that a Fighter seems fairly well positioned for the style of “charge everything and full attack until it dies”. If the Fighter PC uses a different weapon from others you could also probably make a pretty big difference by dropping a bane weapon of the appropriate sort (and perhaps metal) which works against some common type of enemy.
Having played a PC with Paladin levels in Kingmaker I can tell you that there are many, many fights where Smite Evil won’t be a factor and the Paladin might struggle a little to do damage. I’d guess that in those encounters the Fighter’s damage output should look pretty good. It also might be nice if the Fighter decides to be more tactical and go for combat maneuvers since the other PCs aren’t likely to have enough feats to do a lot with them. I guess you could suggest that to the player as something which might be fun. Kingmaker seems like a decent AP for maneuvers.
@SiouL - Forcing the Paladin to lose a “fixed” fight against an NPC swordsman so that the Fighter can try to beat him seems kind of wrong to me. The fact that the swordsman “gets” a “girl PC in your party” makes it a little twisted too. I know that “winner gets the girl” has been a common trope in some stories, but it doesn’t seem like the most enlightened idea, and making “the girl” one of the PCs seems unlikely to go over well (unless maybe that PC’s player likes the idea of some oppression for her PC to break free from - perhaps she’d have a Judith and Holofernes moment…)