I actually think it's likely that Braixen will evolve. It's Serena's ace Pokemon, there have been a couple Hoenn-Kalos parallels so this could be another (like May evolved her Combusken near the end of Advanced), Braixen has been in many battles (including the aerial fight in the TF finale), and the line "Pikachu and Braixen both give it their all" makes much more sense if Braixen evolves. After all, Braixen isn't exactly a difficult opponent for Pikachu, so if Braixen gives it her all and evolves, then Pikachu will also have to put in more effort (and give his all) against a newly evolved Delphox (especially since newly-evolved Pokemon in the anime seem to get a temporary power boost upon evolution).
The only real thing stopping Braixen from evolving is the fact that it's more popular than Delphox in the real world. But this is very close to the end of XYZ so they might let it happen anyway.
I highly doubt the battle will go fully though. There's almost no way Pikachu can lose (imagine the uproar lol) even against a Delphox (though it may certainly have to put in a lot of effort to defeat a fully-evolved starter, especially if Braixen/Delphox replaces scratch with Mystic Fire or Psychic upon evolution), and a loss for Serena wouldn't really drive home the message that the writers nor Ash want to give her. I'd imagine it'd be a tough fight, then Team Rocket interrupts, then Serena and Ash tag team against Team Rocket (like they did in XYZ010), which would be a better representation of their relationship then a fight in which Serena loses.
Another possibility is Braixen evolving while fighting Team Rocket, but that IMHO would be more anti-climactic then having it evolve against Pikachu.
Ultimately though, this battle is a brilliant thing to have in Serena's departure episode. One of the best aspects of her character was seeing her grow from a shy, nervous girl who barely knew what to do in battles without Ash or someone telling her to someone who's confident, is independent, manages her Pokemon on her own without anyone else's help, proactively goes to help others (like Mairin), etc. So a Serena who's confident enough to fight Ash and give everything she's got in the battle is a huge improvement (and shows she has had huge character development) from the shy, nervous girl at the beginning who wouldn't have dared to challenge Ash. In other words, this battle will show how far she's come, and it will be a touching send-off for her character (not to mention for Braixen/Delphox as well, as Fennekin was pretty shy and inexperienced at the beginning of XY as well).
