You should be able to handle most exceptions in your script yourself by using try-catch blocks around the code passages in question. I use XLog in conjunction with the general error handling of JS (try-catch) to control error and general output from my scripts: [Helper: XLog (versatile logging approach for DO scripting))
It allows you to fine tune script output from verbose to none, externally from a button or when used from inside the script (or its config). You also have the choice to mute/increase output for all the scripts that make use of XLog or for specific ones only (iirc).
What you cannot prevent from being visible in the DO output pane is when your script has severe syntax/parsing errors and the script engine is not able to parse it (happens quite easily after editing and saving it externally), but that's the only situation where you are out of control about what is visible from/about your script in the output pane.