My cat too is a fragile guy and has always hated being put in his carrier, going in the car, and being at the vet's, and he's even badly shy with all strangers, so I was very apprehensive about all the many strangers who would be handling him during what amounted to probably a total of 16 hours travel time overall for him. But he came through the whole international trip just fine with a very quick recovery.
It's advised to put a pad or even newspaper or something absorbent at the bottom of the carrier. I think I put towels (he eats pads), then his donut bed. I was interested to find that when i got him home, none of these items seem to have been peed on -- I have no idea if he just didn't pee at all or if ARC at Gatwick cleaned it all out; but even the furry donut bed seemed not to smell -- I washed it anyway. I still to this day have no idea if he just held it the entire time!
Both my cat and the carrier interior were in pristine condition even though he had spent three hours on a drive to the airport then a nine and a half hour flight, out for inspection at ARC, then a couple hours drive to the home destination. He did show distress on the car journey home from Gatwick -- started panting like a dog ( he has always hated car rides).
But when I got the carrier into the living room and let him out to explore his new surroundings, he sprang back to his normal self within a couple of hours, surprisingly. He had such an interesting time checking out the place that he seemed to forget all about his long journey and all the strangers. It is a lot to put them through but they really do seem to bounce back and leave it all behind, even the nervous ones like my cat who is naturally timid but got very settled and happy again very fast.
It's possible that even on a long flight, they just kind of hold it in, pee and poo wise. The same way that you or I often are able to postpone a pee if something starts happening that is heightening our adrenal response, this seems to happen to some travelling pets too. If mine did mess his carrier there was no evidence or at the very least it was cleaned up very well.