How to remove diacritics in computed persisted columns? COLLATE is non-deterministic and cannot be used
Why not just create the table column with a case-insensitive, accent-insensitive collation? This prevents duplicates according to the collation rules, and allows the sort of searches you seem to require:

CREATE TABLE Test
(
    col1 varchar(30) COLLATE SQL_Latin1_General_CP1_CI_AI PRIMARY KEY
);

-- Success
INSERT dbo.Test VALUES ('Gagné');

-- Failed, duplicate key
INSERT dbo.Test VALUES ('Gagne');

-- Success
SELECT * 
FROM dbo.Test 
WHERE col1 = 'gaGNE';


As an important side note, I should mention that T-SQL scalar and multi-statement functions have very poor performance characteristics. Using them in a computed column definition is even worse. Avoid.