Technically insurance coverage follows the vehicle so you should be filing a claim with the company that insures the vehicle you were driving. If you attempt to file a claim with your personal insurance company, they are going to investigate the vehicles involved and that would quickly lead to them denying the claim.
Most homeowners insurance policies cap their jewelry well below the value of your engagement ring. In this case, it's not about raising the value of your personal property but rather getting specific coverage for your ring.
We have published a State of Insurance report that shows average rates for various scenarios. You can use that guide as a general rule of thumb, but ultimately the rates available to you will be specific to your information.