It depends on the insurance company if you need to be listed or not. Some will require the titled/registered owner be on the policy along with the named insured (your fiancee) others will not. Just depends on the underwriters.
Do you have your own insurance someplace else on your car? The letters the BMV sends out is just checking that the licensed driver has at least liability insurance. They are not looking for a specific car (that is why people with no cars get the random letter asking for proof in the mail). If you have a license in Ohio, even if you don't have a car, you have to have the liability to make you legal.