Allow me to correct what appears to be some inaccurate advice posted by some of the non-Arizona lawyers responding to you. Your UIM coverage applies to damages which exceed the amount of lialbity coverage up to the limit of the UIM coveage you purcahsed. UIM coverage is not limited only to the amount of coverage you pruchased which exceeds the liability limit.
I agree with Mr. Rosenstein. Please keep in mind that filing a lawsuit in an Arizona dog bite case within one year is necessary to preserve the best legal theories and give you the greatest chance of success.
"I'm too busy" is a euphemism for "You do not have a case" when a lawyer does not want to actually give you decent advice and tell you what he or she really thinks. No lawyer is too busy to take a case that is worthwhile, otherwise that lawyer would not be talking to you in the first place. Mr. Adams is giving you good adivce. If you look around long enough you may find someone willing to take your case, but I don't think they would be doing you any favors for a whole host of reasons.
As one who regularly handles these kinds of cases, I can emphathize with what a horrible situation this must be for both of you. Criminal prosecution is not a required precursor to civil litigation, but you will want your lawyer to work closely with the prosecutor for a number of reasons, including ensureing the greatest degree of success in any civil action. Many laywers have experience with handling these types of matters with sensitivity and compassion, but you should find someone right away.
In Arizona, insurance follows the vehicle so your corporate policy likely would be primary. Only after that policy is exhausted would the employee's personal policy kick in. Neither insurer could not prusue a subrogation claim against the employee who is an insured under the policies. You corporate carrier will hire a lawyer to represent the employee and perhaps a second lawyer to represent the corporation depending on the allegations that are made in the event a lawsuit is filed.