Are there other ways to dissolve lip filler fast than by injection?

Do you know what kind of filler you had injected? Do you know why the clinic that provided you with lip augmentation is unwilling to reverse it with hyaluronidase? That is quite unusual unless there is another piece to the puzzle not included in your question. As others have stated, the good news is that lip filler generally does not last as long as it does in other areas of the face due to the increased blood flow an movement of the area. This is especially true if it is one of the first times you have had injections. Warm regards.

The only way to get fillers to dissolve is by injecting them with hyaluronidase, assuming you had Hyaluronic acid based fillers. 

Hyaluronic acid fillers can be dissolved using hyaluronidase which has to be injected by a qualified medical professional. Unfortunately, there is not much else that can be done to reduce the amount of lip filler you have.