LUCKIEST 'MAN' ALIVE: All of Longshot's hits are critical hits. If Longshot rolls 12, the attack can't be evaded, damage dealt is increased by an additional 1 and damage can't be reduced below 3.

Would this count as turning the dial for the Phoenix Force dial since rolling doubles triggers it?

Longshot does not roll doubles though. His ability makes it a critical hit. That is not the same thing as rolling two sixes. So no Longshot does not automatically move the Phoenix Resource Dial unless he actually rolls doubles.