Vyxal, 5 bytes

C∑)ṡh

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Explanation

C∑)ṡh # Implicit input
  )ṡ  # Sort by:
C     #  ASCII values
 ∑    #  Sum
    h # First item
      # Implicit output 

Vyxal, 5 4 bytes

‡C∑P

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-1 thanks to @mathcat

Explanation

‡C∑P  # Implicit input
‡  P  # Minimum by:
 C    #  ASCII values
  ∑   #  Sum
      # Implicit output

Old:

C∑)ṡh # Implicit input
  )ṡ  # Sort by:
C     #  ASCII values
 ∑    #  Sum
    h # First item
      # Implicit output