[1] Čuić joined the GNK Dinamo Zagreb Academy in 2015 and rose through the ranks from the club's Under-15 squad to the reserve team by the start of 2018.
Čuić made his debut in the Croatian Second Football League with GNK Dinamo Zagreb II in April 2018, making two appearances in the latter stages of the club's 2017-18 campaign.
Earlier in the year, Čuić helped Dinamo Zagreb's U-23 side advance to the Premier League International Cup final by beating Southampton F.C.
During the club's Under-19 UEFA Youth League appearance in 2016, Čuić started in one game, helping his side claim a 1–0 victory on the road against Italian giants Juventus FC on Dec. 7, 2016.
On January 22, 2020, the Croatian signed, alongside teammate Dominik Rešetar, joined USL Championship side Sporting Kansas City II on loan ahead of the 2020 season.