Ya his office needs to look into it an do any reimbursements. It is essential.
I don't think this proves anything about being dishonest or corrupt. This is a very common problem. Most 'simple-minded' accountants have a 'to-do' list and they follow it blindly. So it is understandable, since its 'common' to be reimbursed from the Gov, and so you send any record of the flights to them. And its common to be reimbursed by PACs. Unless the staffer sits downs and finds out 'which flight is for which' or makes the distinction they are more likely to just look 'oh here's flight on the statement' send it to gov. And when they look at their accounts with the PACs they do the same thing.
Its just incompetence as far as I can see.
Any amount of money that he is 'compensated' for by the gov is not even close to what he returns in unused budget.
Like others have said though, this shows badly on 'management' (President's don't do this shit but still people care), so its important that the campaign clears this out.