I think that would be because 'hoeven' is not a direct verb in this sentence. Although I read on a Flemish website that 'hoeven te' would be okay too here, to my Dutch ears it sounds odd.
The difference is subtle but important to understand: "ik zou niet moeten gaan" is something you say up front when you know it's not smart to go: "I should not go".
"I would not need to go" is something you'd say up front when you don't have to go "Ik zou niet hoeven gaan" (moeten versus hoeven) or in better Dutch: "ik hoef niet te gaan".
Similarly to what tenaaq said, I translated this to "I would not need to go". Is there a way to say that using 'moeten' rather than 'hoeven'?
No, 'moeten' implies more of an obligation. If you use 'need to' you use 'hoeven' in dutch.