If you prefer to leave the car with your ex, if it was me I would get my name on the V5 and I would also insure the vehicle in my name with the ex as a named driver.
You will have to declare to the insurance company where the vehicle will be kept overnight etc.
IMHO I would also get a legally binding contract drawn up regarding the agreed repayments by a solicitor, however, a DIY contract can be also binding.
This situation may be amicable at the moment, however, I guarantee that it will not remain that way for long.
So, brutal as it may sound, I would take the car back from your ex. and if you want to play Mr nice guy, then I would suggest that you repay her the monies that she has already paid to you.
Clean break, let go, move on and all that, is the way to go as these sort of arrangements always end badly and in tears.