Yes, you're right, is it just short for: 約束は守らないといけない and the meaning is still: you have to keep your promise. Your question why the expression is using negative form, I think this is Japanese style. You'll find many many words using negative form, for instance: 申し訳ありません、そうかもしれない、～わけない. hope this helps

ないとね in ”約束は守らないとね” urge the receiver to agree to the idea he must keep his promise. I think we Japanese use this kind of negative form by putting ourselves his position. Like "what will happen if i break a promise?" So, ”約束は守らないとね” is like "you must keep your promise, mustn't you?" Japanese people don't prefer to use clear expressions. Because using ambiguous words or expressions is virtue. It's thought to be important to maintain harmony between people. The receivers guess the meaning of words from the context. Japanese people are used to that ambiguity. i'm sorry. i mistakenly put the same answer as a comment (^_^; )

ないとね in ”約束は守らないとね” urge the receiver to agree to the idea he must keep his promise. I think we Japanese use this kind of negative form by putting ourselves his position. Like "what will happen if i break a promise?" So, ”約束は守らないとね” is like "you must keep your promise, mustn't you?" Japanese people don't prefer to use clear expressions. Because using ambiguous words or expressions is virtue. It's thought to be important to maintain harmony between people. The receivers guess the meaning of words from the context. Japanese people are used to that ambiguity.