I have a personal injury case that was just settled. I agreed to pay a contingent fee and the lawyer’s out of pocket expenses. One of the expenses is a first class airline ticket from Nashville to Atlanta for a deposition – at the cost of $1500! Can the lawyer charge me for that expense?
You should contest the charge, but at the end of the day what will happen will depend on the exact language of the contract, your persistence, and the lawyer’s conscious.
There is nothing legally wrong with the contract providing that the lawyer gets to fly first class at your expense. If that is what you agreed to do then you are legally on the hook.