Unique index on computed column when contributing columns are already unique

I have a table like this:

CREATE TABLE  dbo.DiscountBarcodeList(  ID int NOT NULL IDENTITY    CONSTRAINT PK_DiscountBarcodeList    PRIMARY KEY CLUSTERED, Discount int NOT NULL    CONSTRAINT FK_DiscountBarcodeList_Discount    FOREIGN KEY REFERENCES dbo.Discount (ID), CodeNumber int NOT NULL, AssignedEntity int NULL    CONSTRAINT FK_DiscountBarcodeList_AssignedEntity    FOREIGN KEY REFERENCES dbo.Entity (ID)    ON DELETE SET NULL, BarcodeID AS    CONVERT(      char(10)    , CAST(Discount AS binary(2)) + CAST(CodeNumber AS binary(3))    , 2), CONSTRAINT UQ_DiscountBarcodeList_DiscountCodeNumber    UNIQUE NONCLUSTERED     (      Discount  ASC    , CodeNumber ASC    ));

The table will hold CodeNumbers allocated for Discounts beforehand and assigned to Entitys on demand. Though not on the database level, the values of Discount and CodeNumber will be limited to 2 bytes and 3 bytes respectively. Those limitations, together with the unique constraint on (Discount, CodeNumber), will effectively make the generated BarcodeID values unique as well.

The way this table is supposed to be used is, the application will be passing a @BarcodeID to look up an @Entity to assign to it. If the lookup is successful, then either the row's Entity will be set to @Entity, or the application will be notified that the @BarcodeID is already taken, something along these lines:

BEGIN TRANSACTION;UPDATESET  @OldEntity = Entity, Entity = @EntityWHERE  BarcodeID = @BarcodeID;IF @OldEntity IS NULLBEGIN  COMMIT TRANSACTION;  ...  /* report success */ENDELSEBEGIN  ROLLBACK TRANSACTION;  ...  /* report failure */END;

Now I would like to make BarcodeID sargable. Since I know that the column will only have unique values, I am considering to make the index unique as I think that can make my lookup more efficient. On the other hand, I am concerned that the generated values will have to be checked for uniqueness, which is redundant here since the uniqueness is already guaranteed.

Is it possible to somehow tell whether the benefits, if any, of a unique index on a computed column are going to outweigh the probable overhead of the (unnecessary in this case) uniqueness check? Or at least is it possible to determine that for a scenario like mine? Or am I just overthinking this?