Export from LUTly, install, apply. Three steps and one setting that matters.
Export at 65 point. CapCut samples LUTs coarsely; 65 point avoids banding on smooth gradients.
In desktop CapCut: Adjustment > LUT > Import. Your .cube files appear as thumbnails.
Apply to your original clip, or add as an adjustment layer over several clips.
Seeing colour banding? That is CapCut's LUT sampling, not the file. Re-export from LUTly at 65 point.
CapCut samples LUTs coarsely. Export from LUTly at 65 point and re-import.
No. Custom LUT import is desktop only.
The number of grid points in the LUT. 65 point is a finer grid, which CapCut needs for smooth gradients.