Not having ever worn contacts, I might be out in left field, but...
Can't you just wear them, dump the liquid, and replenish the liquid after arriving at your destination?
My wife wears them - she uses eyedrops because her contacts tend to dry out in flight.
That's entirely feasible, but airplane cabins are VERY dry and have almost no humidity. My eyes dry out easily, and I need saline solution if those contacts are in, which they always are because they correct my vision far better than glasses can.
I have a contact lens case that holds the plastic lens box and a teensy little bottle for saline solution - it holds literally about 30 drops. Of salt water.
But I can't bring it on now.