If you are at fault in a fender bender and your car is insured but not registered will there be a problem?

If the vehicle is not properly registered the insurance company is not liable. Unless at the time the "temporary tag" was valid.

I used to work as an attorney for Liberty Mutual. In California, whether or not a car is currently registered is not relevant to the insurance coverage. The insurance company is still on the hook, even if the registration has expired at some point after the insurance company issued its policy.