iMovie does export 1080p 60fps at 60fps. For that to happen, the project frame rate must be set at 60fps. The resolution and frame rate of a newly created project (one that has never had any clips in it) is determined by the resolution and frame rate of the very first video clip (not photo) placed into the project timeline. There is no other way to set the resolution or frame rate. So, if a 1080p 60fps clip is the first clip added to the newly created project then the project resolution and frame rate will be 1080p 60fps, and it will export at that if you select the 1080 option. This is true even if the first clip is deleted, or moved elsewhere in the timeline, and even of other clips with different frame rates and resolutions are added. Once the project frame rate is set, it cannot be changed. The only way to change it is to create a new project and add a 1080p 60fps clip as the first clip added.
Possibly your project had been set at 30 fps, and thus it would export at that frame rate. If that's the case, then the solution would be to create a new project and set the frame rate as described above.
If the above situation is not your case, then try deleting preferences, as that can solve export issues. To delete preferences, open iMovie while holding down the Option and Command keys and select to delete preferences in the box that appears. iMovie will open in a new library. Reopen your old library to get back to your projects. Now see if you can export at 1080p 60fps. Deleting preferences is a safe procedure that will not cause data loss or disruption to your project.
-- Rich