The causes of action are not different because you have a non-admitted insurer. You potentially have a breach of contract, bad faith and B&P section 17200 claim. Misreading an arbitration award unreasonably to deny a claim would likely amount to insurance bad faith but all circumstances would have to be reviewed. You will need to have an experienced insurance coverage attorney review this. Some firms, like mine, work on an hourly or contingency fee basis.
Breach of contract, breach of the implied covenant of good faith and fair dealing (I.e. Bad faith) are among the causes of action you may be able to raise. You need an attorney that practices in the are of insurance coverage. The attorney would need to see the arbitration award, copies of all applicable policies and any correspondence from your insurance carrier with respect to the denied claims. Many of us offer a free initial consult if you would like to call.