APL(Dyalog Unicode), 91 bytes ^SBCS

{1≥n←≢⍵:0⋄+/(⍵{(×l)++/∧\⊃=/⍺⍵↑⍨¨l←⍺⌊⍥≢⍵}¨(⍵,⊂⍬)[r⍳{⍵+m=(m,2)[r⍳⍵]}⍣≡r←⍋⍋⍵]),∇¨⍵⊂⍨≠m←n<2×⍳n}

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A recursive function computing the total number of comparisons. Rough breakdown:

1≥n←≢⍵:0. If there is only a single string in the list, no comparisons are needed.

∇¨⍵⊂⍨≠m←n<2×⍳n. Split the list into two halves, call the function recursively.

⍵ ... (⍵,⊂⍬)[r⍳{⍵+m=(m,2)[r⍳⍵]}⍣≡r←⍋⍋⍵]. Match up each string with the next larger string in the other half.

{(×l)++/∧\⊃=/⍺⍵↑⍨¨l←⍺⌊⍥≢⍵}¨ For each pairing, count the necessary character comparisons to sort.

APL(Dyalog Unicode), 95 bytes ^SBCS

There is probably some better way to implement the grouping function, would be 24 bytes shorter if the input length was a power of 2.

{+/1+{+/∧\⊃=/⍺⍵↑⍨¨⍺⌊⍥≢⍵}/⍵[⍋⍵][↑⍸(××≠⍤1)k×⍉∧\×k←⊃⍤⍸¨∘.≠⍨↓↑({1=⍵:0⋄⊃,/1 2,¨¨∇¨(⌊,⌈)⍵÷2}≢⍵)[⍋⍵]]}

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