You shouldn’t need to shut anything down. You also shouldn’t need to remove the LUN from the hosts either.
On many occasions I have extended LUNs which were presented to active hosts running SQL Server without stopping the database much less removing the LUN from the host. Unless the EVA has a requirement to do this (which it shouldn’t) there shouldn’t be any reason to not extend the LUN while Exchange is running.
Simply tell the SAN to extend the LUN.
Launch diskpart and rescan on the node which is controlling the LUN.
After diskpart finishes the rescan, extend the volume to fill the drive.