I've done the Cheshire Cat for a few years now. Don't worry about stopping and taking a break or pushing some or all of the climb. There will be a lot of people on the route going up the hill. Some will be resting, some will be pushing, some will be riding slowly and wobbling about, and some will be riding up there fairly quickly.
Everybody's focus will be on their own efforts. Nobody will be thinking about those who choose to walk or rest. Don't be embarrassed. Don't be self conscious. Just get yourself to the top and know that the rest of the middle distance route is flat or downhill from the top of Mow Cop.
I walked the first year, got a lot of miles in training last year and managed to get up it. I've not done as many miles training this year, so I don't think I'll be able to get up this year, but I'm still going to give it a try. But if I have to walk, then I have to walk.