Irish booze Mike put in to change non-alcoholic cocktail (6,4)

'cocktail' is the definition.
(I know that Virgin Mary is a type of Bloody Mary)

'irish booze mike put in to change' is the wordplay.
'irish' becomes 'IR' (abbreviation).
'booze' becomes 'gin' (gin is a kind of booze).
'mike' becomes 'M' (phonetic alphabet: alpha, bravo, charlie etc.).
'put in' indicates putting letters inside (back).
'to change' becomes 'vary' ('vary' can be a synonym of 'change').
'ir'+'gin'+'m'='irginm'
'irginm' inserted inside 'vary' is 'VIRGIN MARY'.

'non-alcoholic' is the link.
I am not very happy about this link. It may belong to another bit of the clue.

Can you help me to learn more?