Well,you can design anything you want, but if you don't pay attention to the restrictions you have, it just wont work.
I would suggest to you to split that DLL in to two DLLs - one that hold the non Qt dependent functionality, and another which is Qt DLL.
This way you can have the non Qt DLL still offer functionality to non Qt applications.

Another option, which I am not sure if it has other problems, is that applications using your DLL will use its QApplication.
You might try declaring your QApplication variable as external.
The problem with this approach is, that Qt application that will need to link to this DLL will have to be designed specifically to work that way.