https://www.corsair.com/us/en/Categories/Products/Cases/Mid-Tower-ATX-Cases/Carbide-Series-SPEC-05-Mid-Tower-Gaming-Case-—-Black/p/CC-9011138-WW
I believe you've made a misinterpretation. There's a picture further down the page that shows fan and radiator compatibility further down the page. Fans in blue bars and radiators in brown.
That's 25mm thick fans PLUS a 27mm thick radiator. This chassis was designed with only fans up top.
Mount the radiator at the front... unless you need the drive cage? If so, yeah, you're kinda boned there.
No, you don't have to buy a new chassis. Send the cooler back and get an air cooler - you may have some options there, but I'd need to know what is being cooled first.