Is it not possible to tell them you don't want to renew, then take out a new policy with the same company as a new customers?
If you go on a comparison site and run your details, most of the time your own insurance company will beat their own quote anyway.
For example, this year (March) my renew quote was £1,800 with Admiral. I had paid 900 the year before, so double. Went on a comparison site, ran my details and Admiral came back with £800. Gave them a ring, the guy on the phone then ran my details and told me he could knock a further £100 off. So I ended up paying £700, £1,100 less than my renew.
Essentially, renewal quotes are nonsense and just a punt in the hope people will auto-renew.