Performance impact of Splitting Table across files
The problem we are facing is from the network team who backup and
  transfer the MDF file over to our DR site every night.


Are they really transferring the mdf file to the DR site? If so, stop, now.

If thankfully backups are being taken, or you can persuade them to start taking backups instead of shifting the mdf, split the backup to multiple files.

BACKUP DATABASE [MyDatabase] 
TO 
    DISK = 'D:\MyDatabase_file1.bak'
    , DISK = 'D:\MyDatabase_file2.bak' 
    , DISK = 'D:\MyDatabase_file3.bak' 
    , DISK = 'D:\MyDatabase_file4.bak' 
WITH
    NAME = 'MyDatabase backup'
    , STATS = 10



  They are also saying it would make it easier to manage allocating
  space for future growth from our virtual infrastructure that it is
  sitting on.


If you want to add additional storage to the database by way of additional files, that is also possible.

ALTER DATABASE [MyDatabase] 
ADD FILE 
(
    NAME = MyDatabase_file2,
    FILENAME = 'D:\MSSQL\Data\MyDatabase_file2.ndf',
    SIZE = 64GB,
    FILEGROWTH = 8GB
)


For this particular situation however, one does wonder why the administrators can't manage storage by way of the (storage) virtualisation layer.