> insist that a renter...have valid insurance [or pay a higher fee]
If you don't own a car it is just about impossible to buy a liability insurance policy. I know this. I tried. I tried very hard.
I don't own a car myself, but since I often rent a car when I travel, I decided that I'd try to buy a third-party liability policy instead of paying the absurd prices that Avis or Dollar charge (>$20 a day).
I called about 10-15 insurance brokers, and not one of them had ever written a "non-owner" policy. The standard answer I got was, "You have to own a car to get liability insurance; yes, it'll cover you for other vehicles you drive, but you have to have your own car for us to create a policy for you".
I realize that you are in the UK, so the situation might be different there. But in the US (and Canada), I think that RelayRides would have to be the one to provide the coverage, because the average RelayRides user (who likely lives in the same city and doesn't own a car) won't have his own liability insurance (and couldn't get it even if he wanted it!).