As long as the people don't have to pay for yet another legislative  legal fishing expedition (like we did for the Medicaid expansion lawsuit folly) than let the good Senator proceed.   However, how can he not call the permanent fund expenditure an appropriation subject to veto?  The State Constitution (Article 9, Section 16) clearly  states that the "appropriation"  for the Alaska Permanent Fund payments is not limited as are other appropriations.  That seems to indicate that the permanent fund is indeed an "appropriation" regardless of the amount  to be appropriated.  That same Constitution (Article 2 Section 15)  states that the Governor may: "....by veto strike or reduce items in appropriation bills".   
Remember, we are about to spend another $800,000 for a study  on how we can become more  efficient with  state spending.  I wonder if legislative lawsuits against the the state government will be looked at?