Yes, Stakeout does double damage upon activation of a Red Card. In this test replay, I had Scyther use Aerial Ace to activate Mabosstiff's Red Card to bring Sandaconda in, making Crunch deal Stakeout damage. On the next turn, Crunch did normal damage without the Stakeout boost. This is proven by these calcs:
252 Atk Stakeout Mabosstiff Crunch vs. 252 HP / 252+ Def Sandaconda: 153-180 (43.9 - 51.7%) -- 10.2% chance to 2HKO
252 Atk Mabosstiff Crunch vs. 252 HP / 252+ Def Sandaconda: 76-91 (21.8 - 26.1%) -- 6.8% chance to 4HKO
Hope this helps.