After winning an arbitration with a homeowners' insurance company, can I sue to recoup costs?

Question Details: Our home was damaged in spring. The adjuster lowballed our damages and we requested a second adjustment. Since it was a small insurance company they only work with one adjustor and so the same adjuster came out to give us the "second opinion". Surprise, the adjustment didn't change. It was a low low ball offer. We ended up going to arbitration incurring about $2000 in costs, not to mention time/stress/etc. We won the arbitration. Can we sue the insurance company to recoup our costs of having to go this route in the first place? The final judgment was significantly more than what they said appraised it as.

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