On a c-clip like the ChryCo 8.25 you can't pull the axles without the clips being out - and you can't get the clips out without pushing the axle into the carrier (which means the shaft comes out or rotates) - since the ChryCo uses a round shaft... rotating doesn't do much...
You'll find that you'll be able to pull the pinion shaft and drop the side gears wil little effort -- on the higher (numerical) ratio rings, starting at like 4.56:1, there's usually a need to shave a tooth to get the assembly back together, so that cut should already be made if you're in that range...