As a youngster, Barbet represented AS Mouliets, Olympique Saint-Pey and FC Vallée de Gamage, before joining the youth setup at Ligue 1 club Bordeaux in July 2007.

[4] He finally won his maiden call into the first team squad for a Europa League group stage match versus Maccabi Tel Aviv on 12 December 2013, but remained an unused substitute during the 1–0 defeat.

[7] Initially signed as a back-up player, Barbet was given his chance at Niort after first-choice central defender Frédéric Bong suffered a long-term injury.

[10] He made his debut with a start in a 4–0 League Cup first round drubbing at the hands of Oxford United on 11 August and possibly put through his own net for the Us' second goal, which was credited to Danny Hylton.

[11][12] With Jack O'Connell largely preferred as third-choice centre back behind Harlee Dean and James Tarkowski,[13] Barbet spent long periods in and out of the matchday squad through to the end of 2015.

[14] The unavailability of Tarkowski saw Barbet make his first appearance in over four months against Preston North End on 23 January 2016 and he assisted Alan Judge for one of Brentford's three goals in the victory.

[34] On 28 September 2019, Barbet received a red card for committing a last-man foul on Hal Robson-Kanu in a 2–0 loss against West Brom meaning he had to serve a one-game suspension.

[38] Barbet helped QPR go six undefeated after his return from injury meaning the team was only six points from playoffs before the league was put on hold due to COVID-19 pandemic.

[41] In the later part of the season, Barbet played in a back five alongside defenders Jordy de Wijs and Rob Dickie which lead to a good run of form.

[46] Barbet was capped by the France national under-18 football team on one occasion, playing 45 minutes in a 2–2 friendly draw with Belgium on 27 January 2011.