The actor that portrayed Barry, Omar Sy was a personal pick of director Colin Trevorrow who'd felt that he was under-used in X-Men and deserved a bigger role. It's implied that his (French) wife may also have played a part in getting this (French) actor into an on-screen role.
Ironically, a lot of Sy's performance seems to have ultimately ended up being cut out of the final script and/or simply left on the cutting-room floor.
"It turns out that I was in the first (Jurassic) because Colin
Trevorrow really wanted to work with me. It was the desire of the
director. There's nothing better then that, if the Director wants you.
If one is imposed [forced] and the director does not want you, its
Omar Sy hints he may not return in Jurassic World 2!
“I wrote the role especially for Omar, I’m a huge fan, I spent a lot of time in France, my wife is French. He has a bigger role than you think – he works for Chris Pratt and is also a Raptor tamer. When Chris, who’s in charge, is done with training the Velociraptors, Omar brings them to their cage and takes care of them, checking their teeth, these kind of things. He has a very special relationship with them. There’s a great scene with Vincent D’Onofrio when Omar speaks French to the Raptors and Vincent pretends to understand what he’s saying. I have not tried to Americanize him – he is French and has a great action scene. Everything he does results in a close relationship with the Velociraptors.”
Director of ‘Jurassic World’ Reveals Omar Sy’s Role in the Film + Whether His Character Dies
In short, his role was essentially at the gift of the original director, his character was peripheral (and evidently disposable) and since Trevorrow didn't direct the second film, there was apparently nobody pushing for Sy to be in the sequel from the studio.
In-universe there's no description of his fate beyond being rescued from the island.