There isn't a way to tell Maplex to terminate the callout at the edge of the symbol or a specified distance from the point symbol. However, using a Composite Callout may provide a workaround for your situation.
The Composite callout is similar to the Balloon callout but has some additional features. One of those is to split the callout line between a balloon-style "dart" and a simple leader line style. Set the Leader line symbol color to null, and adjust the Proportion of dart drawn as leader to control how far the invisible leader portion will extend.
This produces and end result like this.
As for the balloon overlapping other features - add a feature weight to the point label class to instruct Maplex to avoid placing labels over the top of the point features.