If you are faithfully relaying to us the true meaning of what these RIM support techs are saying to you, then I'm afraid that you have been unfortunate enough as to speak to two complete morons. The assertion that you must use a different BBID with each distinct device is flatly incorrect. Repeat: THESE MORONS AT RIM SUPPORT ARE WRONG. As others have pointed out in this thread, App World even has an option available to filter the app list to show only those apps installed on the current device, so that those with multiple devices sharing the same BBID don't have to slog through their phone's apps when managing the apps on their PlayBook, and vice versa. If you call back, you might want to ask them why this feature is there, if you're not supposed to do it that way?
The problem you're having seems to be an issue with the specific app. You should contact support for that app. It may be that they simply tell you that they don't license the app cross platform, and you will have to choose one either your phone or your tablet on which to install this app.