A closed no payment claim will follow you to the next carrier, but only some carriers use them in rating. In this situation, you should have started with the at-fault auto coverage and not filed the homeowners claim. Now you know for the future. If you file for the repair through your homeoweners policy, you should make sure they know the auto involved and get the insurance company. Your company will then "subrogate" (or get the money they paid to fix your house from the other carrier). You then have a subrogated claim. Subrogated claims look better than closed no payment claims.
As to specific advice to your situation, I'm not sure why you don't have coverage up to the damage amount unless you are seriously underinsured. At this point, as much as I hate to suggest it, I think your best bet is to talk to a lawyer. The thing to keep in mind here is that the person that damaged your home is likely liable for the damage. Neither your limit of insurance nor their limit of insurance mitigates the actual liability.
If you want to know what other carriers see when you switch carriers, order a CLUE property report on yourself. I think you can still get one free per year. LexisNexis provides these to carriers as the primary source of past claim activity.