So, my question is, do I have grounds to get out of my insurance without paying those fees?
Yes. You've set out your grounds here. You are entitled to dispute any demand for money. It would be worthwhile, if you are going to dispute this matter, to put the situation in writing and to propose some solutions other than simply not paying (for example, paying pro-rata for the policy, waiting for the new licence, etc.). If you do decide to dispute the matter, you need to be prepared for threatening letters and, possibly, debt collectors and court action if a fair solution can't be found.
It is important that you are careful not to make promises like "the licence will be here next week" but you should be honest about the situation.