Hey do yall think had Ned Stark went forward with Little Finger's suggestion of taking the iron throne and becoming king and protector of the realm and exposing the Lannisters that Little Finger would've actually backed him in the throne room instead of betraying him?
That's just been one thing where I've been thinking is this cuz he's just a sneaky little bastard that can't be trusted or did he have a possible game plan had that went down? (Yeah I know it doesn't really matter much right now).
Ned could never play the game nor did he want the throne. Littlefinger doesn't want the throne either he just wants to manipulate those on and around the throne to elevate his status and economic gains.
So to answer your question no Littlefinger would not have betrayed him immediately but if Ned ever did something to spurn Littlefinger or if Littlefinger could get him out of the way to get Cat in his arms he would do it.