Ring insurance?

I use USAA. You don't have to prove that it's lost, it's integrity based. I pay $4 a month for mine - check your normal insurance carrier. Usually the rates are very inexpensive if you already have policies!

Our rings are all insured through Jewelers Mutual - they are very affordable. I've never had to submit a claim through them, so I'm not quite sure what their process is, but check them out!

My ring is insured through our housing insurance. Luckily never filed a claim, but have known people who did. Just read your policy carefully. Most don't protect against you losing it, so when you file you would say something like "Took it off to wash my hands, and then it wasn't there when I went to put it back on" rather than "I lost it"

We added ours to our homeowners insurance

I also use Jewelers. It came recommended from the jeweler who designed my ring. I haven't had to file a claim, but it seemed pretty straight forward.

I added mine to my homeowners policy. You will have to submit the appraisal from your jeweler. That is how they will quote you a price to add. My ring is about $220 a year to insure.
You don't have to prove it was lost or stolen it is on the honor system.
If it was damaged you might have to send it to them as proof .