Can a Pokemon with an attack that does damage to the opponent's Bench get through Mr. Mime's "Bench Barrier" Ability if the Attacking Pokemon has Stealthy Hood attached?
No, it does not get through. The Bench Barrier Ability is not "being done" to the Attacking Pokemon.
Source: TPCi Rules Team (2019-05-30)

Relevant cards

Note: All applicable cards may not be listed below.
G1_52
SM10_186
SM10_186