It is usually better to file it with your insurance company if the amount of claim is significantly more than your deductible. "Significant" is a relative term, for each person's situation. The purpose of car insurance is to make you whole in this situation. Any affect that it has on your future rate is out-weighed by the out of pocket expense.

Even small claims are sometimes better to file with your carrier, because small claims don't hit most carriers' claim thresholds. The amount of a claim is a factor or a reason to factor it into your rate.

It would appear in your situation that you would benefit from filing the claim. State insurance laws can differ regarding this issue but yes, filing a claim with your own insurance company can allow insurance companies to raise your rates in most states. Usually though this type of claim does not raise your premiums unless you already have a number of claims.