When I worked at a wildlife sanctuary I had to cut up wallaby carcasses with a meat cleaver. I was totally vegan at the time, and at first I watched someone else do it, and thought I was going to be sick, but after a while I just ignored where it came from and what it was, and saw it solely as Devil and Quoll food. That didn't make getting your face splattered in blood and guts any more fun, but it helped me push my veganism to the side temporarily. I still hated it though, and I much preferred making the veg feeds over the meat feeds...
Funnily, I found it easier to deal with meat that still had the fur and guts, as opposed to the meat that didn't... I guess seeing the fur and body parts helped me in knowing that the animal wasn't cruelly farmed and killed for a selfish purpose. It was a wild wallaby, that was unfortunate enough to live on land where humans didn't want it.
I also had to put dead day-old chicks into the feeds, and meal worms. Washing up containers and seeing raw and cooked meat doesn't sound like it would be hard, based on experiencing the above.
What would upset me, though, is knowing that the animal was farmed and murdered just so its flesh could sit on a plate and be eaten by greedy people. The animals at the sanctuary needed the meat, but the people in the restaurant didn't!