IRL application a day after BRP expiry

Submitted an IRL application a day after my BRP expired but in theory before the 30 month period ended as indicated on the visa letter.
Will this mean I have over stayed and I should expect a refusal?
Why did I wait so long to apply in the first place, failed the Life in the UK test and there were limited spots to redo the test.
What are my options if I get refused?
Can I write a cover letter to explain the situation?

If you make a late application, which is subject to 39E, then they expect a good reason. Did you give one?

I did not, but I still have the opportunity to upload documents, so I guess a cover letter making my case is needed. A good reason will be hard to make are there any good examples of not so good reasons getting through? (The good reasons stated are very specific and evidence bound which I don’t have)

Basically a lot of drama with the Life in the U.K. test some my fault some the company that’s runs it. In the end I submitted it with out a life in the U.K. pass but I did upload the evidence of it afterwards.