You can do this, it just takes a bit of downtime.

Here are the steps:

Begin by provisioning the new LUNs and presenting them to the cluster. Create disk resources around them and bring them into the SQL group.

Shut SQL Server down, but leave the clustered disk resources online.

Copy everything exactly from Old disk(s) to New Disk(s).

Take the old disks out of the resource group.

Change the old disk drive letters to some unused letters.

Change the new disk drive letters to the old ones.

Change the SQL Server group dependencies so that the SQL Server service is dependent on each disk resource.

Restart SQL.

Check everything before deleting the old disk resources and removing the LUNs from the cluster.

SQL only uses drive paths to locate data files. If you change things "under the covers" so that new disks have the same folder structure and drive letters as the old ones, then SQL won't notice the difference.