The only thing on my 05 Journey with an audible alarm that chimes is the jack down warning. All the others (so far) are some form of electronic sounding buzz or beep.
My first reaction is you have a intermittent hot short that's causing voltage to be applied to the hot side of whatever device creates the chime sound. That device is probably at the end of it's own circuit like a radio speaker would be in relationship to a radio. If this is the case then to behave the way you describe the hot short would have occurring somewhere along the hot (not black) wire leading to that device, probably near it. Based on your description the hot source would have to be coming from something in the chassis electrical system that is only activated when the ignition is on and maybe only under certain conditions.
Do you ever smell the "hot wire smell" when this occurs? That would be a sure sign of a short. Lack of a smell means it's more subtle and might mean the problem, including a short, internal to a sealed component.
If your up to it you can try making heads or tails of the Winnie electrical diagrams
http://www.winnebagoind.com/diagram/...001_Wiring.htm and consider getting under the dash to trace some wires. You might find all you need is some electrical tape.
If it comes down to going to a dealer then I suggest to help minimize dealership hours you and call Winnebago Service Administration at (800)-537-1885 with your coach serial number and ask for advice including whether there are any TSB's that might fit your symptoms.