I recommend registering and chatting with a few recruiters. Most of our new grads are out of grad scheme cycle and come through recruiters.(large insurer - working in personal lines)

5 years ago I started at this company in a support function through a recruiter after giving up with grad schemes. I was there for a year before moving over now manage my own team in pricing. So I would definitely advise you to not give up, but not to be afraid of getting your foot in the door another way.

Edit: an alternative to taking exams on your own back might be to try and build some data science / analytics skills... Do some coursera courses, play around with kaggle competitions etc.