Any thoughts on how to remove this faucet / handle
It may be a Kohler or at least have the same design. Try this :


  1  Hold on to the handle while turning the handle base counterclockwise by hand or with a strap wrench, taking care to protect the finish.
  
  2  Remove any remaining components to expose the valve. Set all parts aside for re-use.
  
  3  The valve can now be replaced or other repairs made.
  
  4  Reassemble handle parts in reverse order.


https://support.kohler.com/hc/en-us/articles/360002095334-Remove-the-Handles-for-the-Finial-Bathroom-or-Kitchen-Faucet