Basically BBK is spot on here.
V5 purely shows who is legally responsible for taxing the vehicle, nothing else. It matters not a jot who's name is on the top, but they're the person who will be responsible for filling out any S172s for speeding etc, and more importantly who can be taken to court and pursued for any civil parking invoices by PPCs.
You already own the vehicle, as you paid for it. Your receipt proves that, or money transfer to the dealer or whatever. That's your legal proof, right there. The arrangement you have with your ex is a different contract: They're buying the car from you at the rate of £12K over 36 months, and this is what you need a contract for. Easy to do, even something like "Mrs X agrees to pay Mr Y £333.33 per month for 36 months for BMW X5 registration XX00 XXX, commencing from xx/xx/xx. The vehicle shall not become the property of Mrs X until the amount is paid in full" and signed & dated by both is more than sufficient.
If your Ex doesn't own the car, some insurers may be funny. However, that's not your problem so it's not something I'd worry about. She'll find someone to insure her, she may just pay a bit extra for the benefit. HOWEVER, personally I would be insuring it myself and getting her to pay me. Why? Well, what if she under insures it, or forgets to pay the insurance, or otherwise messes it up? If the car is then written off by her, you lose your asset and have no claim other than chasing her directly for the money which is going to prove troublesome I suspect. No, far better to insure it yourself and then you know it's done.