*In a Double battle.
I use Regigigas and my Audino with ability Healer and Entrainment move.
If I use Entrainment into Regigigas would fail or Regigigas get rid of Slow Start?*
Yes it should.This is the entrainmet definition.
Entrainment makes the target's ability the same as the user's.
It does not work on Pokémon with the abilities Truant, Multitype or Zen Mode
Source:Pokémon DB Slow Start
Not much more to say. I just tested it and I have the replay right here.
Hope I helped.